{"id":110298,"date":"2026-09-26T14:25:11","date_gmt":"2026-09-26T06:25:11","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/110298.html"},"modified":"2026-09-26T14:25:11","modified_gmt":"2026-09-26T06:25:11","slug":"%cf%84-%ce%b7-%e7%a6%bb%e6%95%a3%e5%9c%ba-%c2%b7-%e7%ac%ac%e4%b8%89%e5%8d%b7","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/110298.html","title":{"rendered":"\u03c4\u2013\u03b7 \u79bb\u6563\u573a \u00b7 \u7b2c\u4e09\u5377"},"content":{"rendered":"<h3>\u6574\u9664\u56fe\u7684\u4e09\u5c42\u8c31\u5730\u8c8c\u3001\u539f\u5b50\u5854\u4e0e \u03b6 \u8272\u6563\u2014\u2014\u79bb\u6563\u2192\u8fde\u7eed\u6781\u9650\u7684\u65e0\u9ed1\u7bb1\u62a5\u544a&#xff08;v1.0&#xff09;<\/h3>\n<hr \/>\n<h3>0. \u9605\u8bfb\u89c4\u5219<\/h3>\n<p>0.1 \u6807\u6ce8\u4f53\u7cfb&#xff08;\u627f\u5377\u4e8c&#xff09;<br \/>\n[\u53ef\u8bc1] \u672c\u5377\u5185\u7ed9\u51fa\u5b8c\u6574\u63a8\u5bfc&#xff0c;\u65e0\u201c\u663e\u7136\u201d\u5b57\u6837 \/ [\u771f\u8dd1] \u5b9e\u9645\u6267\u884c&#xff0c;\u9644 N\u3001\u5bb9\u5dee\u3001\u5b9e\u9a8c\u7f16\u53f7 \/ [\u731c\u60f3] \u6709\u8bc1\u636e\u672a\u8bc1\u660e \/ [\u8bc1\u4f2a] \u88ab\u5b9e\u9a8c\u6740\u6b7b \/ [\u672a\u590d\u6838] \u6709\u4e3b\u5f20\u65e0\u672c\u8f6e\u5b9e\u9a8c \/ [\u5f00\u653e] \u65e0\u653b\u51fb\u8def\u5f84\u7684\u60ac\u6848\u3002<br \/>\n0.2 \u7ea6\u5b9a\u9501\u5b9a&#xff08;\u672c\u5377\u4e00\u5207\u6570\u5b57\u7684\u552f\u4e00\u53e3\u5f84&#xff09;<\/p>\n<ul>\n<li>\u56fe G_N&#xff1a;\u9876\u70b9 {2,\u2026,N}&#xff0c;\u8fb9 \u27fa \u6574\u9664&#xff08;\u627f\u5377\u4e00&#xff09;<\/li>\n<li>GCC_N&#xff1a;\u542b\u9876\u70b9 2 \u7684\u8fde\u901a\u5206\u91cf&#xff08;\u5b58\u5728\u6027\u552f\u4e00\u6027 &#061; \u5377\u4e8c\u5b9a\u7406 5&#xff09;<\/li>\n<li>\u4e24\u4e2a\u7b97\u5b50\u4e0d\u53ef\u6df7\u6dc6&#xff1a;\u539f\u59cb L &#061; D \u2212 A&#xff1b;\u5f52\u4e00\u5316 \u2112 &#061; I \u2212 D<sup>{\u22121\/2}AD<\/sup>{\u22121\/2}&#xff0c;\u5747\u5728 GCC_N \u4e0a<\/li>\n<li>\u03bb1^raw \u4e0e \u03bb1^norm \u5206\u522b\u6307\u4e24\u7b97\u5b50\u7684\u6700\u5c0f\u6b63\u7279\u5f81\u503c<\/li>\n<li>\u8ba1\u6570\u7a97\u53e3 \u00b110\u207b6&#xff08;\u4f9d\u636e\u89c1 \u00a75 \u6848\u4f8b 1&#xff09;&#xff1b;\u03c0(x) \u7d20\u6570\u8ba1\u6570&#xff1b;k_j :&#061; \u03c0(N\/j) \u2212 \u03c0(N\/(j&#043;1))&#xff0c;\u5373\u533a\u95f4 (N\/(j&#043;1), N\/j] \u5185\u7d20\u6570\u4e2a\u6570<\/li>\n<li>\u5bb6\u65cf F_p^{(j)} :&#061; {mp : 1 \u2264 m \u2264 j}&#xff0c;p \u53d6\u81ea band_j<br \/>\n0.3 \u65e0\u9ed1\u7bb1\u7684\u64cd\u4f5c\u5b9a\u4e49&#xff1a;\u56db\u95ee\u534f\u8bae<br \/>\n\u6bcf\u4e2a\u6570\u503c\u4e3b\u5f20\u5fc5\u987b\u56db\u95ee\u5168\u7b54&#xff0c;\u7f3a\u4e00\u5373\u9ed1\u7bb1&#xff1a;<br \/>\n(i) \u5b9a\u4e49\u2014\u2014\u54ea\u4e2a\u7b97\u5b50\u3001\u54ea\u4e2a\u56fe\u3001\u54ea\u4e2a\u7a97\u53e3&#xff1f;<br \/>\n(ii) \u5b9e\u9a8c\u2014\u2014\u54ea\u6bb5\u4ee3\u7801\u3001\u54ea\u4e2a N\u3001\u54ea\u4e2a\u5bb9\u5dee&#xff1f;<br \/>\n(iii) \u5224\u636e\u2014\u2014\u9884\u8bbe\u7684\u901a\u8fc7\/\u5931\u8d25\u6761\u4ef6\u662f\u4ec0\u4e48&#xff08;\u5148\u4e8e\u7ed3\u679c\u6ce8\u518c&#xff09;&#xff1f;<br \/>\n(iv) \u8bc1\u4f2a\u8def\u5f84\u2014\u2014\u4ec0\u4e48\u89c2\u6d4b\u4f1a\u6740\u6b7b\u5b83&#xff1f;<br \/>\n0.4 \u8303\u56f4<br \/>\n\u672c\u5377\u53ea\u5199\u6570\u5b66\u5c42\u3002\u03c4\u2013\u03b7\u3001\u534f\u540c\u672c\u4f53\u8bba\u3001\u2207\u03b7\u3001\u4f5c\u7528\u91cf\u53cd\u95ee\u9898\u4e0d\u8fdb\u5165\u6b63\u6587\u2014\u2014\u5bf9\u4e0b\u8ff0\u4efb\u4f55\u4e3b\u5f20\u5b83\u4eec\u65e2\u975e\u5fc5\u8981\u4ea6\u4e0d\u53ef\u68c0\u9a8c&#xff0c;\u5265\u79bb\u540e\u65e0\u635f\u3002<\/li>\n<\/ul>\n<hr \/>\n<h3>1. \u4efb\u52a1\u4e0e\u4e3b\u7ed3\u679c<\/h3>\n<p>\u5377\u4e00\u3001\u5377\u4e8c\u786e\u7acb\u4e86\u7cbe\u786e\u7ec4\u5408\u7ed3\u6784&#xff08;\u5ea6\u516c\u5f0f\u3001\u5206\u91cf\u3001\u76f4\u5f84&#xff09;\u4e0e\u7edf\u8ba1\u6e10\u8fd1&#xff08;\u03b6 \u7ba1\u77e9\u3001\u03b3 \u7ba1 log \u4fee\u6b63&#xff09;\u3002\u5377\u4e8c\u9057\u7559\u4e24\u4e2a\u672a\u95ed\u5408&#xff1a;\u03bb\u221e &#061; \u03c02\/48&#xff08;\u731c\u60f3&#xff09;&#xff0c;\u4ee5\u53ca\u201c\u79bb\u6563\u2192\u8fde\u7eed\u201d\u8fd9\u53e5\u8bdd\u7684\u6570\u5b66\u542b\u4e49\u3002\u672c\u5377\u56de\u7b54\u540e\u8005&#xff0c;\u5e76\u7ed9 \u03bb\u221e \u4e00\u4e2a\u65b0\u7684\u7ed3\u6784\u5730\u5740\u3002<br \/>\n\u4e3b\u7ed3\u679c&#xff08;\u5148\u9884\u89c8&#xff0c;\u00a72\u2013\u00a74 \u9010\u6761\u8bc1\u660e&#xff09;&#xff1a;<\/p>\n<p>\u5f52\u4e00\u5316\u8c31\u7684\u5e95\u90e8\u662f\u4e09\u5c42\u5730\u8c8c&#xff1a;L0 \u6536\u655b\u7684\u79bb\u57df\u4f53\u5c42&#xff08;\u627f\u8f7d \u03bb\u221e&#xff09;&#xff1b;L1 \u79bb\u57df\u68af\u5c42&#xff08;hub\u2194band \u5171\u632f&#xff09;&#xff1b;L2 \u88ab\u7cbe\u786e\u9489\u6b7b\u7684\u7b97\u672f\u539f\u5b50\u5854\u2014\u2014\u5176\u4e00\u822c\u5f62\u5f0f\u662f\u672c\u5377\u65b0\u5b9a\u7406&#xff1a;\u539f\u5b50 &#061; \u81ea\u76f8\u4f3c gadget \u7684 Dirichlet \u8c31&#xff0c;\u91cd\u6570 &#061; k_j \u2212 1\u3002\u8fde\u7eed\u6781\u9650\u4f4f\u5728\u79bb\u57df\u5c42&#xff0c;\u5176\u8272\u6563\u5173\u7cfb\u662f \u03b6&#xff1b;\u79bb\u6563\u4ee5\u96f6\u6d4b\u5ea6\u4ee3\u6570\u539f\u5b50\u7684\u5f62\u5f0f\u6c38\u5b58\u3002\u534a\u5f20\u8c31\u662f\u7b97\u672f\u7684&#xff08;\u547d\u9898 C&#xff09;\u3002<\/p>\n<hr \/>\n<h3>2. \u539f\u5b50\u5854&#xff1a;\u5c01\u95ed\u6027\u5f15\u7406\u4e0e\u56db\u6761\u5b9a\u7406<\/h3>\n<h4>2.0 \u903b\u8f91\u8d77\u70b9&#xff1a;\u4e3a\u4ec0\u4e48\u4f1a\u60f3\u5230\u201c\u5854\u201d<\/h4>\n<p>\u5377\u4e8c\u5b9a\u7406 6 \u7684 1\/\u03c62 \u662f\u4e00\u4e2a\u6ca1\u88ab\u89e3\u91ca\u7684\u7cbe\u786e\u6570\u3002\u8ffd\u95ee\u201c\u4e3a\u4ec0\u4e48\u7cbe\u786e\u201d&#xff0c;\u7b54\u6848\u53ea\u80fd\u662f&#xff1a;\u5b58\u5728\u4e00\u65cf\u652f\u6491\u5c01\u95ed\u7684\u5b50\u7ed3\u6784&#xff0c;\u5176\u5185\u90e8\u7279\u5f81\u65b9\u7a0b\u4e0e\u5168\u56fe\u89e3\u8026\u3002\u628a\u201c\u54ea\u4e9b\u9876\u70b9\u96c6\u7684\u90bb\u57df\u5c01\u95ed\u4e8e\u81ea\u8eab&#043;\u5c11\u6570\u951a\u70b9\u201d\u95ee\u5230\u5e95&#xff0c;\u5c31\u540c\u65f6\u5f97\u5230\u81ea\u76f8\u4f3c&#xff08;\u5f15\u7406 0&#xff09;\u3001\u5c01\u95ed\u6027&#xff08;\u5f15\u7406 1&#xff09;\u548c\u6574\u5ea7\u5854&#xff08;\u5b9a\u7406 2\/3&#xff09;\u3002<\/p>\n<h4>2.1 \u5f15\u7406 0&#xff08;\u81ea\u76f8\u4f3c&#xff09;\u3000[\u53ef\u8bc1 &#043; \u771f\u8dd1 E1]<\/h4>\n<p>\u9648\u8ff0&#xff1a;\u8bbe a \u2265 2&#xff0c;M &#061; \u230aN\/a\u230b\u3002G_N \u5728 {a, 2a, \u2026, Ma} \u4e0a\u7684\u8bf1\u5bfc\u5b50\u56fe\u540c\u6784\u4e8e\u542b 1 \u7684\u9664\u6570\u56fe G*M&#xff08;\u9876\u70b9 {1,\u2026,M}&#xff0c;\u8fb9 u\u2014v \u27fa u|v&#xff09;&#xff0c;\u540c\u6784\u6620\u5c04 am \u21a6 m\u3002<br \/>\n\u8bc1\u660e&#xff1a;\u8bf1\u5bfc\u8fb9 (am_i, am_j) \u5b58\u5728 \u27fa am_i | am_j \u27fa m_i | m_j\u3002\u220e<br \/>\n\u9a8c\u8bc1 E1&#xff1a;G500 \u9650 2 \u500d\u6570\u3001G750 \u9650 3 \u500d\u6570\u3001G1250 \u9650 5 \u500d\u6570\u2014\u2014\u4e09\u4e2a\u4e0d\u540c\u7a97\u53e3\u7ed9\u51fa\u540c\u4e00\u4e2a\u56fe G250&#xff1a;\u8fb9\u96c6\u9010\u4e00\u76f8\u7b49&#xff0c;\u5168\u8c31\u5dee 0.0\u3002<br \/>\n\u610f\u4e49&#xff1a;hub a \u7684\u90bb\u57df\u5c31\u662f G{\u230aN\/a\u230b}\u2014\u2014\u6bcf\u4e2a\u9876\u70b9\u662f\u81ea\u5df1\u5b87\u5b99\u91cc\u7684 1\u3002\u5377\u4e8c \u00a70.1 \u5bf9 1 \u7684\u6392\u9664\u5728\u8fd9\u91cc\u5f97\u5230\u56de\u62a5&#xff1a;1 \u6ca1\u6709\u88ab\u5220\u9664&#xff0c;\u800c\u662f\u88ab\u5206\u5e03\u4e86\u3002\u6b64\u5f15\u7406\u662f \u00a74.4 \u4e0d\u52a8\u70b9\u9012\u5f52\u4e0e \u00a72.2 gadget \u7ed3\u6784\u7684\u5171\u540c\u6e90\u5934\u3002<\/p>\n<h4>2.2 \u5f15\u7406 1&#xff08;\u5bb6\u65cf\u5c01\u95ed\u6027\u2014\u2014\u672c\u5377\u7684\u6280\u672f\u6838\u5fc3&#xff09;\u3000[\u53ef\u8bc1]<\/h4>\n<p>\u9648\u8ff0&#xff1a;\u8bbe j \u2265 2 \u6ee1\u8db3 j(j&#043;1) &lt; N&#xff08;\u5e72\u51c0\u533a\u95f4&#xff09;&#xff0c;p \u2208 band_j\u3002\u5219 F_p^{(j)} \u4e2d\u4efb\u4e00\u9876\u70b9 mp \u7684\u90bb\u5c45\u6ee1\u8db3<br \/>\n<span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">N(mp)\u2286Fp(j)\u222a{2,3,\u2026,j}N(mp) \\\\subseteq F_p^{(j)} \\\\cup \\\\{2, 3, \\\\dots, j\\\\}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mord mathnormal\">p<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2286<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.3211em;vertical-align: -0.3831em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em\"><span class=\"\" style=\"top: -2.453em;margin-left: -0.1389em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">p<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3831em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u222a<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">{<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">3<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span><\/span><\/span><br \/>\n\u8bc1\u660e&#xff08;\u4e09\u6b65&#xff0c;\u5168\u90e8\u521d\u7b49&#xff09;&#xff1a;<br \/>\n\u2460 p \u7684\u90bb\u5c45 &#061; \u5176\u500d\u6570 {2p, \u2026, \u230aN\/p\u230bp}\u3002\u56e0 p \u2208 (N\/(j&#043;1), N\/j]&#xff0c;\u6709 \u230aN\/p\u230b &#061; j&#xff0c;\u6545\u5168\u90e8\u843d\u5728 F \u5185&#xff1b;p \u65e0\u56e0\u5b50\u90bb\u5c45\u3002<br \/>\n\u2461 mp \u7684\u500d\u6570&#xff1a;(mp)k \u2264 N \u27fa mk \u2264 \u230aN\/p\u230b &#061; j&#xff0c;\u6545 mk \u2264 j&#xff0c;\u500d\u6570\u5168\u5728 F \u5185\u3002<br \/>\n\u2462 mp \u7684\u56e0\u5b50 &#061; {d : d | m} \u222a {pd\u2019 : d\u2019 | m}&#xff08;p \u2224 m \u7531 p &gt; j \u4fdd\u8bc1&#xff0c;\u800c p &gt; j \u7531 j(j&#043;1) &lt; N \u4fdd\u8bc1&#xff09;\u3002\u540e\u8005 \u2264 pd \u2264 pj \u2264 N \u4e14\u5728 F \u5185&#xff1b;\u524d\u8005\u662f\u5c0f\u6574\u6570&#xff0c;\u843d\u5728 {2,\u2026,j}\u3002\u220e<br \/>\n\u610f\u4e49&#xff1a;\u6bcf\u4e2a band_j \u7d20\u6570 p \u5f20\u51fa\u4e00\u4e2a gadget\u2014\u2014\u6309\u5f15\u7406 0&#xff0c;gadget \u7684\u5185\u90e8\u7ed3\u6784\u5c31\u662f G*_j&#xff0c;\u5176\u6bcf\u4e2a\u9876\u70b9 m&#xff08;2 \u2264 m \u2264 j&#xff09;\u88ab\u951a\u5230\u5168\u56fe\u7684\u5c0f\u6574\u6570 m \u4e0a\u3002\u539f\u5b50\u5c42\u7684\u5168\u90e8\u73b0\u8c61 &#061; \u8fd9\u4e9b gadget \u7684 Dirichlet \u8c31&#xff08;\u951a\u70b9\u53d6\u96f6&#xff09;\u3002<\/p>\n<h4>2.3 \u5b9a\u7406 2&#xff08;\u539f\u59cb\u7b97\u5b50\u539f\u5b50\u5854&#xff09;\u3000[\u53ef\u8bc1&#xff08;\u6784\u9020&#xff09;&#043; \u771f\u8dd1&#xff08;\u7b49\u53f7&#xff0c;j&#061;2,3&#xff09;]<\/h4>\n<p>\u9648\u8ff0&#xff1a;\u5bf9\u6bcf\u4e2a\u5e72\u51c0 j \u4e0e G*j-\u5e26\u951a gadget \u7684\u6bcf\u4e2a Dirichlet \u7279\u5f81\u503c \u03bc&#xff08;&#061; L \u5728 F_p \u4e0a\u7684 j\u00d7j \u4e3b\u5b50\u9635\u7684\u7279\u5f81\u503c&#xff0c;\u4e0d\u4f9d\u8d56 p\u2014\u2014\u56e0 \u230aN\/(pm)\u230b \u4e0e d(m) \u5728\u5e26\u5185\u4e3a\u5e38\u6570&#xff09;&#xff0c;\u03bc \u662f L(GCC_N) \u7684\u7cbe\u786e\u7279\u5f81\u503c&#xff0c;\u91cd\u6570 \u2265 k_j \u2212 1\u3002<br \/>\n\u8bc1\u660e&#xff1a;\u53d6\u5411\u91cf x&#xff1a;\u5728 \u222a_p F_p \u4e0a&#xff0c;\u6bcf\u4e2a F_p \u5185\u53d6 gadget \u7684 Dirichlet \u672c\u5f81\u77e2 v&#xff08;\u951a\u70b9\u503c\u4e3a\u96f6&#xff09;&#xff0c;\u7cfb\u6570 c_p&#xff1b;\u951a\u70b9 {2,\u2026,j} \u53ca\u5176\u4f59\u9876\u70b9\u53d6\u96f6\u3002\u7531\u5f15\u7406 1&#xff0c;\u7279\u5f81\u65b9\u7a0b\u53ea\u5728\u4e09\u7c7b\u9876\u70b9\u975e\u5e73\u51e1&#xff1a; \u5185\u70b9\u2014\u2014\u5373 gadget \u65b9\u7a0b&#xff0c;\u81ea\u52a8\u6ee1\u8db3&#xff1b; \u951a\u70b9 m\u2014\u2014(Lx)m &#061; d_m\u00b70 \u2212 \u03a3{u\u2208N(m)} x_u&#xff0c;\u800c N(m) \u2229 supp(x) &#061; {mp : p \u2208 band_j}&#xff0c;\u4e14 d{mp} \u5728\u5e26\u5185\u5e38\u6570&#xff0c;\u65b9\u7a0b\u9000\u5316\u4e3a v_{m}\u00b7\u03a3c_p\u00b7w &#061; 0&#xff0c;\u5373\u6bcf\u4e2a\u951a\u70b9\u7ed9\u51fa\u540c\u4e00\u4e2a\u7ea6\u675f \u03a3c_p &#061; 0&#xff1b; \u5176\u4f59\u9876\u70b9\u2014\u2014\u90bb\u57df\u4e0e\u652f\u6491\u4e0d\u76f8\u4ea4&#xff0c;\u81ea\u52a8\u6ee1\u8db3\u3002k_j \u4e2a\u7cfb\u6570\u51cf 1 \u4e2a\u7ea6\u675f&#xff0c;\u5f97 k_j \u2212 1 \u7ef4\u3002\u220e<br \/>\n\u5df2\u9a8c\u8bc1\u5b9e\u4f8b&#xff1a;<\/p>\n<table>\n<tr>gadgetDirichlet \u503c\u91cd\u6570\u5f8b\u9a8c\u8bc1<\/tr>\n<tbody>\n<tr>\n<td>j&#061;2&#xff08;\u8fb9&#xff0c;\u951a{2}&#xff09;<\/td>\n<td>(3\u2212\u221a5)\/2 &#061; 1\/\u03c62&#xff0c;(3&#043;\u221a5)\/2<\/td>\n<td>k2\u22121<\/td>\n<td>N \u2208 [26,150] \u7a77\u5c3d&#xff08;125 \u70b9\u65e0\u4e00\u4f8b\u5916&#xff09;&#043; 200\/500\/1000\/2000\/5000&#xff1b;\u7279\u5f81\u5411\u91cf\u53d6\u8bc1&#xff1a;\u652f\u6491 &#061; 30 \u4e2a\u60ac\u6302\u9876\u70b9\u3001x2 &#061; 0.0\u3001x_{2p}\/x_p &#061; 0.618034<\/td>\n<\/tr>\n<tr>\n<td>j&#061;3&#xff08;\u8def\u5f84 2p\u2014p\u20143p&#xff0c;\u951a{2,3}&#xff09;<\/td>\n<td>2\u2212\u221a2&#xff0c;2&#xff0c;2&#043;\u221a2<\/td>\n<td>k3\u22121<\/td>\n<td>N&#061;100\/200\/500\/1000&#xff1a;mult(2\u2212\u221a2) &#061; 13 &#061; k3\u22121&#xff08;N&#061;1000&#xff09;\u7b49<\/td>\n<\/tr>\n<tr>\n<td>j \u2265 4&#xff1a;\u6784\u9020\u4e0e\u5f15\u7406\u540c\u6837\u8986\u76d6 [\u53ef\u8bc1]&#xff1b;\u7b49\u53f7\u7684\u6570\u503c\u786e\u8bc1&#xff08;\u00b110\u207b6 \u8ba1\u6570 &#043; \u6b8b\u5dee 10\u207b15 \u6784\u9020&#xff09;\u539f\u5b9a\u672c\u8f6e\u6267\u884c&#xff0c;\u5de5\u5177\u914d\u989d\u8017\u5c3d&#xff0c;\u767b\u8bb0\u4e3a[\u672a\u590d\u6838]\u3002<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>\u5bf9\u5377\u4e8c\u5b9a\u7406 6 \u7684\u4fee\u6b63&#xff08;\u627f\u524d\u8f6e&#xff09;&#xff1a;\u591a\u91cd\u6570\u5f8b\u65e0\u4f8b\u5916&#xff1b;\u88ab\u4f8b\u5916\u6253\u7834\u7684\u662f\u95f4\u9699\u9648\u8ff0\u3002\u7a77\u5c3d\u9a8c\u8bc1 [26,150]&#xff1a;\u03bb1^raw &#061; 1\/\u03c62 \u5931\u8d25\u4e8e {26\u201329, 33, 34, 35, 39, 57}\u3002\u4e24\u7c7b\u673a\u5236&#xff1a;k2 &#061; 1&#xff08;N&#061;33, 57\u2014\u2014\u5e26\u5185\u4ec5\u4e00\u7d20\u6570&#xff0c;\u65e0\u76f8\u6d88\u5bf9\u8c61&#xff0c;\u91cd\u6570\u4e3a 0&#xff0c;\u5b9e\u6d4b N&#061;57 gap &#061; 0.383017&#xff09;&#xff1b;k2 \u2265 2 \u4f46\u4ed6\u6a21\u8dcc\u7834&#xff08;26\u201329, 34, 35, 39&#xff0c;\u6765\u6e90\u672a\u5206\u7c7b[\u5f00\u653e]&#xff09;\u3002\u731c\u60f3&#xff1a;57 \u4e4b\u540e\u65e0\u4f8b\u5916[\u731c\u60f3]\u3002<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>2.4 \u5b9a\u7406 3&#xff08;\u5f52\u4e00\u5316\u7b97\u5b50\u539f\u5b50\u5854\u2014\u2014\u672c\u5377\u4e2d\u5fc3\u7ed3\u679c&#xff09;\u3000[\u53ef\u8bc1&#xff08;\u6784\u9020&#xff09;&#043; \u771f\u8dd1&#xff08;\u7b49\u53f7&#xff0c;j&#061;2,3&#xff09;]<\/h4>\n<p>\u9648\u8ff0&#xff1a;\u4e0e\u5b9a\u7406 2 \u5e73\u884c&#xff0c;gadget \u6362\u4e3a \u2112 \u7684\u4e3b\u5b50\u9635\u3002\u4e00\u822c\u5f8b&#xff1a;\u4e0d\u4e0e\u5176\u4ed6 gadget \u5171\u4eab\u7684\u503c&#xff0c;\u91cd\u6570 \u2265 k_j \u2212 1&#xff1b;\u5171\u4eab\u503c\u91cd\u6570 &#061; \u03a3k_j \u2212&#xff08;\u951a\u70b9\u7ea6\u675f\u7cfb\u7684\u79e9&#xff09;\u3002<br \/>\nj&#061;2\/3 \u5171\u4eab\u7684\u673a\u5236&#xff08;\u4e3a\u4ec0\u4e48\u6070\u597d\u662f 1\u22121\/\u221a2&#xff09;&#xff1a;j&#061;2 gadget \u662f\u5355\u8fb9\u60ac\u6302&#xff0c;\u5185\u70b9\u65b9\u7a0b (1\u2212\u03bb)x_p &#061; x_{2p}\/\u221a2 \u4e0e (1\u2212\u03bb)x_{2p} &#061; x_p\/\u221a2 \u7ed9 (1\u2212\u03bb)2 &#061; 1\/2\u3002j&#061;3 \u7684\u5bf9\u79f0\u6a21&#xff1a;\u4e2d\u5fc3 p \u6302\u4e24\u6761\u6743 1\/2 \u7684\u817f&#xff0c;\u603b\u8026\u5408 2\u00b7(1\/2)2 &#061; 1\/2\u2014\u2014\u603b\u951a\u5b9a\u8026\u5408\u76f8\u7b49 \u27f9 \u8c31\u5171\u4eab\u3002\u6545 1\u22121\/\u221a2 \u4e0e 1&#043;1\/\u221a2 \u540c\u65f6\u5c5e\u4e8e\u4e24\u4e2a gadget\u3002<br \/>\n\u7ea6\u675f\u8ba1\u6570&#xff08;\u4e0e\u5b9e\u6d4b\u7cbe\u786e\u543b\u5408&#xff09;&#xff1a;\u5171\u4eab\u503c\u4e0a&#xff0c;\u672a\u77e5\u91cf k2 &#043; k3 \u4e2a\u7cfb\u6570&#xff1b;\u951a 2 \u7684\u65b9\u7a0b \u03a3a &#043; \u03a3b\/\u221a2 &#061; 0&#xff0c;\u951a 3 \u7684\u65b9\u7a0b \u03a3b &#061; 0&#xff08;j&#061;3 \u72ec\u5360\u951a 3&#xff09;\u3002\u540e\u8005\u4ee3\u5165\u524d\u8005\u5f97 \u03a3a &#061; 0\u3002\u4e24\u4e2a\u72ec\u7acb\u7ea6\u675f \u27f9 \u91cd\u6570 &#061; k2 &#043; k3 \u2212 2\u3002\u671b\u8fdc\u955c\u6052\u7b49\u5f0f k2 &#043; k3 &#061; \u03c0(N\/2) \u2212 \u03c0(N\/4) \u7ed9\u51fa\u5c01\u95ed\u5f62\u5f0f&#xff1a;<br \/>\n<span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u00a0mult(1\u221212)&#061;mult(1&#043;12)&#061;\u03c0\u2009\u2063(N2)\u2212\u03c0\u2009\u2063(N4)\u22122\u00a0\\\\boxed{\\\\ \\\\text{mult}\\\\left(1 &#8211; \\\\tfrac{1}{\\\\sqrt 2}\\\\right) &#061; \\\\text{mult}\\\\left(1 &#043; \\\\tfrac{1}{\\\\sqrt 2}\\\\right) &#061; \\\\pi\\\\!\\\\left(\\\\tfrac{N}{2}\\\\right) &#8211; \\\\pi\\\\!\\\\left(\\\\tfrac{N}{4}\\\\right) &#8211; 2\\\\ }<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 2.48em;vertical-align: -0.99em\"><\/span><span class=\"mord\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.49em\"><span class=\"\" style=\"top: -4.48em\"><span class=\"pstrut\" style=\"height: 4.48em\"><\/span><span class=\"boxpad\"><span class=\"mord\"><span class=\"mord\"><span class=\"mspace\">\u00a0<\/span><span class=\"mord text\"><span class=\"mord\">mult<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size2\">(<\/span><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8451em\"><span class=\"\" style=\"top: -2.551em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord sqrt mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9128em\"><span class=\"svg-align\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mtight\" style=\"padding-left: 0.833em\">2<\/span><\/span><span class=\"\" style=\"top: -2.8728em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"hide-tail mtight\" style=\"min-width: 0.853em;height: 1.08em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1272em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.394em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.538em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size2\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord text\"><span class=\"mord\">mult<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size2\">(<\/span><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8451em\"><span class=\"\" style=\"top: -2.551em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord sqrt mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9128em\"><span class=\"svg-align\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mtight\" style=\"padding-left: 0.833em\">2<\/span><\/span><span class=\"\" style=\"top: -2.8728em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"hide-tail mtight\" style=\"min-width: 0.853em;height: 1.08em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1272em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.394em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.538em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size2\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c0<\/span><span class=\"mspace\" style=\"margin-right: -0.1667em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8723em\"><span class=\"\" style=\"top: -2.655em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.394em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.345em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c0<\/span><span class=\"mspace\" style=\"margin-right: -0.1667em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8723em\"><span class=\"\" style=\"top: -2.655em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.394em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.345em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\">\u00a0<\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.49em\"><span class=\"pstrut\" style=\"height: 4.48em\"><\/span><span class=\"stretchy fbox\" style=\"height: 2.48em;border-style: solid;border-width: 0.04em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.99em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><br \/>\n\u9a8c\u8bc1&#xff08;\u56db\u7ec4\u72ec\u7acb\u5b9e\u9a8c&#xff0c;N&#061;500\/1000\/2000\/3000&#xff09;&#xff1a;<\/p>\n<table>\n<tr>Nk2k3k2&#043;k3\u22122\u7d27\u7a97\u53e3\u8ba1\u6570&#xff08;\u00b110\u207b6&#xff09;\u7c07\u5185\u6563\u5e03\u4e0a\u539f\u5b50\u8ba1\u6570<\/tr>\n<tbody>\n<tr>\n<td>500<\/td>\n<td>15<\/td>\n<td>8<\/td>\n<td>21<\/td>\n<td>21<\/td>\n<td>\u2014<\/td>\n<td>21<\/td>\n<\/tr>\n<tr>\n<td>1000<\/td>\n<td>28<\/td>\n<td>14<\/td>\n<td>40<\/td>\n<td>40<\/td>\n<td>1.9\u00d710\u207b15<\/td>\n<td>40<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>47<\/td>\n<td>26<\/td>\n<td>71<\/td>\n<td>71<\/td>\n<td>\u2014<\/td>\n<td>71<\/td>\n<\/tr>\n<tr>\n<td>3000<\/td>\n<td>71<\/td>\n<td>36<\/td>\n<td>105<\/td>\n<td>105<\/td>\n<td>\u2014<\/td>\n<td>105<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ul>\n<li>\u76f4\u548c\u5224\u5b9a&#xff08;N&#061;1000&#xff09;&#xff1a;40 \u7ef4\u672c\u5f81\u7a7a\u95f4\u5728\u5e26 2 \u5bb6\u65cf\u9876\u70b9\u4e0a\u7684\u79e9 &#061; 27 &#061; k2\u22121&#xff0c;\u5e26 3 \u4e0a &#061; 13 &#061; k3\u22121&#xff0c;27&#043;13 &#061; 40\u2014\u2014\u7cbe\u786e\u76f4\u548c&#xff0c;\u5e26\u95f4\u96f6\u6df7\u5408\u3002<\/li>\n<li>\u6784\u9020\u6b8b\u5dee&#xff1a;\u5e26 2 \u7ec4\u5408&#xff08;\u03a3a&#061;0&#xff09;\u2016\u2112x \u2212 \u03bbx\u2016\/\u2016x\u2016 &#061; 2.6\u00d710\u207b17&#xff1b;\u5e26 3 \u7ec4\u5408&#xff08;\u03a3b&#061;0&#xff09;&#061; 1.2\u00d710\u207b16\u3002<\/li>\n<li>\u53cd\u6784\u9020&#xff08;\u201c\u5355\u4e00\u5171\u4eab\u7ea6\u675f\u201d\u66ff\u4ee3\u5047\u8bf4&#xff09;&#xff1a;\u53ea\u6ee1\u8db3 \u03a3a &#043; \u03a3b\/\u221a2 &#061; 0\u3001\u03a3b &#061; 3 \u7684\u6df7\u5408\u5411\u91cf&#xff0c;\u6b8b\u5dee 1.2\u00d710\u207b2&#xff0c;\u5931\u8d25\u4f4d\u7f6e\u7cbe\u786e\u5b9a\u4f4d\u4e8e\u951a 3 \u7684\u65b9\u7a0b&#xff08;\u6b8b\u5dee 8.2\u00d710\u207b2&#xff09;\u2014\u2014\u66ff\u4ee3\u5047\u8bf4\u88ab\u6784\u9020\u6027[\u8bc1\u4f2a]\u3002<br \/>\nj \u2265 4 \u5c42&#xff1a;gadget \u7684 Dirichlet \u503c&#xff08;&#061; \u4e3b\u5b50\u9635\u7279\u5f81\u503c&#xff0c;\u524d\u8f6e\u5b9e\u6d4b&#xff0c;\u5e26\u5185\u4e0d\u4f9d\u8d56 p&#xff09;&#xff1a;j&#061;4: 0.2962&#xff1b;j&#061;5: 0.29549&#xff1b;j&#061;6: 0.29669&#xff1b;j&#061;7: 0.29622&#xff1b;j&#061;8: 0.29749&#xff08;\u53e6\u5404\u6709\u8f83\u9ad8\u503c&#xff09;\u3002\u6309\u5b9a\u7406\u9884\u6d4b\u91cd\u6570 &#061; k_j \u2212 1&#xff08;N&#061;1000&#xff1a;6, 7, 3, 3, 0\u2014\u2014k8&#061;1 \u6545\u4e3a 0&#xff09;\u3002\u7b49\u53f7\u9a8c\u8bc1\u672a\u6267\u884c[\u672a\u590d\u6838]&#xff0c;\u4e0d\u5f97\u5f15\u7528\u4e3a\u5df2\u8bc1\u3002\u4e00\u4e2a\u7ed3\u6784\u6027\u6ce8\u8bb0&#xff1a;j \u2265 4 \u7684 gadget \u4e0e\u666e\u901a\u9876\u70b9&#xff08;\u5982\u951a 4&#xff09;\u7ea0\u7f20&#xff0c;\u4e14\u5c0f\u6574\u6570\u951a {2,\u2026,j} \u88ab\u591a\u4e2a\u5e26\u5171\u4eab\u2014\u2014\u5c01\u95ed\u6027\u5f15\u7406\u4ecd\u7136\u8986\u76d6\u6784\u9020&#xff0c;\u4f46\u5171\u4eab\u503c\u7684\u7ea6\u675f\u79e9\u8ba1\u7b97\u66f4\u7e41&#xff0c;\u7b49\u53f7\u9700\u9010\u5e26\u9a8c\u8bc1\u3002<br \/>\nj&#061;3 \u7684\u53cd\u5bf9\u79f0\u6a21\u53e6\u7ed9 \u03bb &#061; 1&#xff08;\u5f52\u4e00\u5316&#xff09;&#xff0c;\u91cd\u6570 \u2265 k3\u22121&#xff0c;\u540c\u6cd5[\u53ef\u8bc1]\u3002<\/li>\n<\/ul>\n<h4>2.5 \u5b9a\u7406 4&#xff08;\u539f\u5b50\u7a00\u91ca\u5f8b&#xff09;\u3000[\u53ef\u8bc1&#xff08;\u6761\u4ef6\u7ed3\u6784\u89c1\u4f9d\u8d56\u94fe&#xff09;&#043; \u771f\u8dd1]<\/h4>\n<p>\u9648\u8ff0&#xff1a;\u539f\u5b50\u5c42&#xff08;\u4ec5\u8ba1 j&#061;2\/3 \u5171\u4eab\u503c&#xff09;\u8c31\u8d28\u91cf \u03bc_N &#061; (\u03c0(N\/2)\u2212\u03c0(N\/4)\u22122)\/|GCC_N| \u6ee1\u8db3<br \/>\n<span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03bcNln\u2061N&#061;14&#043;38ln\u2061N&#043;O\u2009\u2063(1ln\u20612N)\\\\mu_N \\\\ln N &#061; \\\\frac14 &#043; \\\\frac{3}{8\\\\ln N} &#043; O\\\\!\\\\left(\\\\frac{1}{\\\\ln^2 N}\\\\right)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03bc<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0074em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">4<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0074em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">8<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">3<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.4em;vertical-align: -0.95em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mspace\" style=\"margin-right: -0.1667em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.2116em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mop\"><span class=\"mop\">ln<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8984em\"><span class=\"\" style=\"top: -3.1473em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7884em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><br \/>\n\u903b\u8f91\u4e0e\u8bc1\u660e\u8981\u70b9&#xff1a;\u4e00\u9636&#xff1a;\u03c0(N\/2)\u2212\u03c0(N\/4) \u2248 (N\/2)\/ln(N\/2) \u2212 (N\/4)\/ln(N\/4)&#xff0c;\u5c55\u5f00\u76f8\u6d88\u540e &#061; N\/(4 lnN)\u3002\u7cfb\u6570 1\/4 \u7684\u5e72\u51c0\u6027\u6765\u81ea (1&#043;ln2)\/2 \u2212 (1&#043;2ln2)\/4 &#061; 1\/4 \u7cbe\u786e\u6210\u7acb&#xff08;\u56e0 ln4 &#061; 2ln2&#xff09;\u30023\/8 \u6765\u81ea PNT \u7b2c\u4e8c\u9879 x\/ln2x \u7684\u8d21\u732e\u3002|GCC| &#061; N \u2212 1 \u2212 \u03c0(N) &#043; \u03c0(N\/2) \u7684\u5c55\u5f00\u7167\u505a\u3002\u220e<br \/>\n\u771f\u8dd1&#xff08;c(N) :&#061; \u03bc_N lnN&#xff0c;\u6574\u6570\u516c\u5f0f\u8ba1\u7b97&#xff09;&#xff1a;<\/p>\n<table>\n<tr>N104105106107<\/tr>\n<tbody>\n<tr>\n<td>c(N) \u5b9e\u6d4b<\/td>\n<td>0.2926<\/td>\n<td>0.2854<\/td>\n<td>0.2795<\/td>\n<td>0.2753<\/td>\n<\/tr>\n<tr>\n<td>\u9884\u6d4b \u00bc &#043; 3\/(8lnN) \u2212 2lnN\/GCC<\/td>\n<td>0.2887<\/td>\n<td>0.2823<\/td>\n<td>0.2771<\/td>\n<td>0.2733<\/td>\n<\/tr>\n<tr>\n<td>\u6b8b\u5dee<\/td>\n<td>&#043;0.004<\/td>\n<td>&#043;0.003<\/td>\n<td>&#043;0.002<\/td>\n<td>&#043;0.002<\/td>\n<\/tr>\n<tr>\n<td>\u6b8b\u5dee\u5355\u8c03\u6536\u7f29&#xff0c;\u91cf\u7ea7 &#061; O(1\/ln2N) \u9879\u3002\u5c0f N&#xff08;\u22643000&#xff09;c(N) \u6709 \u00b10.008 \u6da8\u843d\u2014\u2014\u6765\u81ea \u03c0 \u7684 O(\u221aN) \u7d20\u6570\u6da8\u843d&#xff0c;\u975e\u6a21\u578b\u9519\u8bef\u3002<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>\u4f9d\u8d56\u94fe&#xff08;\u8bda\u5b9e\u767b\u8bb0&#xff09;&#xff1a;\u03bc_N \u7684\u6574\u6570\u516c\u5f0f\u6761\u4ef6\u4e8e\u5b9a\u7406 3 \u7684\u7b49\u53f7&#xff1b;\u7b49\u53f7\u5df2\u673a\u5668\u9a8c\u8bc1\u81f3 N&#061;3000&#xff1b;N \u2265 104 \u7684 c(N) \u662f\u516c\u5f0f\u8ba1\u7b97&#xff0c;\u8c31\u672a\u8dd1\u3002\u65e0\u6761\u4ef6\u53ef\u8bc1\u7684\u90e8\u5206&#xff1a;\u03bc_N \u2265 (k2&#043;k3\u22122)\/<\/td>\n<td>GCC<\/td>\n<td>\u2192 (1\/4&#043;o(1))\/lnN&#xff08;\u6784\u9020\u4e0b\u754c &#043; PNT&#xff09;\u3002<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>\u524d\u8f6e\u88c1\u5b9a\u8bb0\u5f55&#xff1a;\u5916\u90e8\u590d\u7b97\u8005\u66fe\u4ee5\u201cc_local \u5355\u8c03\u4e0a\u5347\u3001lnlnN \u56de\u5f52\u53d1\u6563\u201d\u64a4\u56de\u6b64\u5f8b\u2014\u2014\u88ab\u62d2\u7edd&#xff0c;\u7406\u7531\u4e09\u6761&#xff1a;c_local \u65e0\u5b9a\u4e49&#xff08;\u4e0d\u662f c(N)&#xff0c;\u65e0 N \u6807\u7b7e&#xff0c;\u65e0\u7a97\u53e3\u534f\u8bae&#xff09;&#xff1b;\u5176\u56de\u5f52\u5185\u90e8\u77db\u76fe&#xff08;\u7aef\u70b9\u53cd\u89e3 N\u22481.3\u00d7105 \u2260 \u58f0\u79f0\u7684 106&#xff09;&#xff1b;2\u20133 \u4e2a\u6570\u91cf\u7ea7\u5185 1\/lnN \u4e0e\u4efb\u4f55\u7f13\u5347\u51fd\u6570\u65e0\u5224\u522b\u529b\u3002\u00bc \u662f\u5b9a\u7406\u7cfb\u6570&#xff0c;\u4e0d\u662f\u88ab\u730e\u7684\u7b2c\u56db\u4e2a\u5e38\u6570\u3002<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>2.6 \u547d\u9898 C&#xff08;\u539f\u5b50\u5360\u6bd4\u2014\u2014\u534a\u5f20\u8c31\u662f\u7b97\u672f\u7684&#xff09;\u3000[\u53ef\u8bc1&#xff08;\u4e0b\u754c&#xff09;&#043; \u542f\u53d1\u5f0f&#xff08;\u6781\u9650&#xff09;&#043; \u771f\u8dd1&#xff08;\u5206\u6570&#xff09;]<\/h4>\n<h3>\u9648\u8ff0&#xff1a;\u8c31\u4e2d\u5c5e\u4e8e\u673a\u5668\u7cbe\u5ea6\u7b80\u5e76\u7c07&#xff08;\u91cd\u6570 \u2265 2 \u7684\u539f\u5b50&#xff09;\u7684\u672c\u5f81\u503c\u5360\u6bd4&#xff0c;\u968f N \u4e0a\u5347&#xff0c;\u6e10\u8fd1\u4e0b\u754c \u2192 1\/2\u3002<br \/>\n\u903b\u8f91&#xff1a;\u6bcf\u4e2a\u5e26 j \u8d21\u732e\u7ea6 j \u4e2a Dirichlet \u503c \u00d7 (k_j\u22121) \u91cd&#xff1b;k_j \u2248 N\/(j(j&#043;1)lnN) \u5bf9 j \u226a \u221aN \u6709\u6548&#xff0c;k_j \u2265 2 \u8981\u6c42 j \u2272 \u221a(N\/lnN)\u3002\u6c42\u548c&#xff1a;<br \/>\n<span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2211j\u2264N\/ln\u2061Nj\u2009(kj\u22121)\u2248Nln\u2061N\u2211j1j&#043;1\u2248N2(1\u22122ln\u2061ln\u2061Nln\u2061N&#043;\u22ef\u2009)\\\\sum_{j \\\\le \\\\sqrt{N\/\\\\ln N}} j\\\\,(k_j &#8211; 1) \\\\approx \\\\frac{N}{\\\\ln N}\\\\sum_{j}\\\\frac{1}{j&#043;1} \\\\approx \\\\frac{N}{2}\\\\left(1 &#8211; \\\\frac{2\\\\ln\\\\ln N}{\\\\ln N} &#043; \\\\cdots\\\\right)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 2.8856em;vertical-align: -1.8356em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05em\"><span class=\"\" style=\"top: -1.6082em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mrel mtight\">\u2264<\/span><span class=\"mord sqrt mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.0369em\"><span class=\"svg-align\" style=\"top: -3.4286em\"><span class=\"pstrut\" style=\"height: 3.4286em\"><\/span><span class=\"mord mtight\" style=\"padding-left: 1.19em\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><span class=\"mord mtight\">\/<\/span><span class=\"mspace mtight\" style=\"margin-right: 0.1952em\"><\/span><span class=\"mop mtight\"><span class=\"mtight\">l<\/span><span class=\"mtight\">n<\/span><\/span><span class=\"mspace mtight\" style=\"margin-right: 0.1952em\"><\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><span class=\"\" style=\"top: -3.0089em\"><span class=\"pstrut\" style=\"height: 3.4286em\"><\/span><span class=\"hide-tail mtight\" style=\"min-width: 0.853em;height: 1.5429em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4197em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8356em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0315em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2248<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.7741em;vertical-align: -1.4138em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3603em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4138em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8804em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2248<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.4em;vertical-align: -0.95em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3603em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3714em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><br \/>\n\u771f\u8dd1&#xff1a;N&#061;1000\/2000 \u5f52\u4e00\u5316\u8c31\u7b80\u5e76\u7c07\u5360\u6bd4&#xff08;\u524d\u8f6e\u5b9e\u6d4b\u7684\u7c07\u7ed3\u6784&#xff1a;40 \u91cd\u3001105 \u91cd\u539f\u5b50\u7b49&#xff09;\u7ea6 30% \/ 35%\u91cf\u7ea7\u2014\u2014\u65b9\u5411\u4e0e\u91cf\u7ea7\u4e0e 1\/2 \u5f8b\u76f8\u5bb9[\u672a\u590d\u6838&#xff1a;\u7cbe\u786e\u5206\u6570\u672c\u8f6e\u672a\u91cd\u7b97]\u3002<br \/>\n\u8fb9\u754c&#xff1a;\u8de8\u5e26\u503c\u5171\u4eab\u4f1a\u4f7f\u7ea6\u675f\u79e9\u4e0a\u5347\u3001\u603b\u91cf\u7565\u964d&#xff1b;\u5171\u4eab\u662f\u5c0f\u4e8f\u635f&#xff08;j&#061;2\/3 \u5171\u4eab\u4e24\u503c\u5404\u635f 0&#xff09;\u3002\u6e10\u8fd1 1\/2 \u662f[\u542f\u53d1\u5f0f]\u3002<\/h3>\n<h3>3. \u8272\u6563\u5b9a\u7406&#xff1a;\u79bb\u6563\u2192\u8fde\u7eed\u7684\u5b9e\u6865<\/h3>\n<h4>3.1 \u5b9a\u7406 5\u3000[\u53ef\u8bc1 &#043; \u771f\u8dd1]<\/h4>\n<p>\u9648\u8ff0&#xff1a;\u52a0\u6743\u6574\u9664\u56fe&#xff08;\u8fb9 (a, am) \u6743 m^{\u2212s}&#xff0c;s &gt; 0&#xff09;\u4e0a&#xff0c;\u5e73\u9762\u6ce2 f(a) &#061; a^{\u2212it} \u7684 Rayleigh \u5546\u6ee1\u8db3\u6052\u7b49\u5f0f<br \/>\n<span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">RN(s,t)&#061;2N\u22121\u2211m&#061;2\u230aN\/2\u230bm\u2212s(1\u2212cos\u2061(tln\u2061m))(\u230aNm\u230b\u22121)\\\\mathcal{R}_N(s,t) &#061; \\\\frac{2}{N-1}\\\\sum_{m&#061;2}^{\\\\lfloor N\/2\\\\rfloor} m^{-s}\\\\big(1-\\\\cos(t\\\\ln m)\\\\big)\\\\left(\\\\Big\\\\lfloor \\\\frac{N}{m}\\\\Big\\\\rfloor &#8211; 1\\\\right)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathcal\">R<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">s<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.2281em;vertical-align: -1.2671em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7693em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.961em\"><span class=\"\" style=\"top: -1.8829em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.386em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">\u230a<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><span class=\"mord mtight\">\/2<\/span><span class=\"mclose mtight\">\u230b<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2671em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">m<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8213em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2212<\/span><span class=\"mord mathnormal mtight\">s<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.4em;vertical-align: -0.95em\"><\/span><span class=\"mop\">cos<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><span class=\"mord\"><span class=\"delimsizing size1\">)<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\"><span class=\"delimsizing size2\">\u230a<\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3603em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">m<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mord\"><span class=\"delimsizing size2\">\u230b<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><br \/>\n\u4e14 N \u2192 \u221e \u65f6<br \/>\n<span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">RN(s,t)\u21922\u2009Re[\u03b6(s&#043;1)\u2212\u03b6(s&#043;1\u2212it)]\\\\mathcal{R}_N(s,t) \\\\to 2\\\\,\\\\mathrm{Re}\\\\big[\\\\zeta(s&#043;1) &#8211; \\\\zeta(s&#043;1 &#8211; it)\\\\big]<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathcal\">R<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">s<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2192<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.2em;vertical-align: -0.35em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathrm\">Re<\/span><\/span><span class=\"mord\"><span class=\"delimsizing size1\">[<\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0738em\">\u03b6<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">s<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0738em\">\u03b6<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">s<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.2em;vertical-align: -0.35em\"><\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mclose\">)<\/span><span class=\"mord\"><span class=\"delimsizing size1\">]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><br \/>\n\u903b\u8f91&#xff08;\u600e\u4e48\u60f3\u5230&#xff09;&#xff1a;\u6574\u9664\u8fb9\u662f\u7b97\u672f\u5e8f\u5217 (a, 2a, 3a, \u2026)\u2014\u2014\u5bf9\u6570\u5750\u6807\u4e0b\u662f\u957f\u5ea6 ln m \u7684\u5e73\u79fb\u8df3\u8dc3&#xff1b;\u4e58\u6cd5\u534a\u7fa4\u7684\u7279\u5f81 a^{\u2212it} \u662f\u5929\u7136\u8bd5\u51fd\u6570&#xff1b;\u5377\u4e8c \u00a75 \u4e94\u6761\u5149\u6ed1\u5316\u5931\u8d25\u5171\u540c\u6307\u5411\u201c\u6781\u9650\u5fc5\u987b\u663e\u5f0f\u4fdd\u7559\u8df3\u8dc3\u7ed3\u6784\u201d&#xff0c;\u800c\u5e73\u9762\u6ce2\u6b63\u662f\u68c0\u9a8c\u8df3\u8dc3\u7ed3\u6784\u7684\u63a2\u9488\u3002<br \/>\n\u8bc1\u660e&#xff08;\u4e94\u884c&#xff0c;\u65e0\u9ed1\u7bb1&#xff09;&#xff1a;\u2460 |f(a) \u2212 f(am)|2 &#061; |a^{\u2212it}|2|1 \u2212 m^{\u2212it}|2 &#061; 2 \u2212 2cos(t ln m)&#xff08;\u521d\u7b49&#xff09;\u3002\u2461 \u6c42\u548c\u8303\u56f4\u4e3a a &#061; 2\u2026N&#xff0c;m &#061; 2\u2026\u230aN\/a\u230b\u3002\u2462 \u4ea4\u6362\u6b21\u5e8f&#xff1a;\u56fa\u5b9a m&#xff0c;\u5408\u6cd5 a \u7684\u4e2a\u6570 &#061; #{a \u2265 2, a \u2264 N\/m} &#061; \u230aN\/m\u230b \u2212 1\u2014\u2014\u6052\u7b49\u5f0f\u6210\u7acb\u3002\u2463 \u6781\u9650&#xff1a;(\u230aN\/m\u230b\u22121)\/(N\u22121) \u2192 1\/m \u9010\u70b9&#xff1b;\u88ab m^{\u2212s\u22121} \u63a7\u5236&#xff08;s &gt; 0&#xff09;&#xff0c;\u63a7\u5236\u6536\u655b\u3002\u2464 1 \u2212 cos \u03b8 &#061; Re[1 \u2212 e^{i\u03b8}] \u628a\u548c\u5f0f\u8ba4\u51fa\u4e3a \u03b6 \u7684\u5dee\u3002\u220e<br \/>\n\u9a8c\u8bc1&#xff1a;<\/p>\n<ul>\n<li>\u6052\u7b49\u5f0f\u5728\u771f\u5b9e\u52a0\u6743\u56fe\u4e0a\u76f4\u63a5\u68c0\u9a8c&#xff08;N&#061;500, s&#061;1, t&#061;0.7 \u4e0e 2.0&#xff09;&#xff1a;\u56fe\u77e9\u9635 Rayleigh vs \u516c\u5f0f&#xff0c;\u5dee 2.2\u00d710\u207b16 \/ 0.0&#xff08;\u673a\u5668\u7cbe\u5ea6&#xff09;\u3002<\/li>\n<li>\u6781\u9650&#xff08;\u5bf9 t&#061;1.3&#xff09;&#xff1a;s&#061;1.5 \u8bef\u5dee 4.0\u00d710\u207b5 \u2192 4.0\u00d710\u207b7&#xff08;N: 105\u2192107&#xff09;&#xff1b;s&#061;0.5 \u8bef\u5dee 3.2\u00d710\u207b2 \u2192 3.1\u00d710\u207b3&#xff1b;s&#061;1, 2 \u5404\u70b9\u5168\u8fc7&#xff1b;mpmath \u03b6 \u72ec\u7acb\u4ea4\u53c9\u9a8c\u8bc1&#xff08;\u4f8b&#xff1a;2Re[\u03b6(2)\u2212\u03b6(2\u22120.7i)] &#061; 0.65243574&#xff09;\u3002<\/li>\n<li>\u8bef\u5dee\u5f8b[\u53ef\u8bc1]&#xff1a;O(N^{\u2212min(s,1)})\u2014\u2014\u8fb9\u754c\u5c42\u4f30\u8ba1&#xff1a;\u03a3_m m^{\u2212s}\u00b7O(m\/N)&#xff1b;s&lt;1 \u65f6 ~ N^{\u2212s}\/(1\u2212s)&#xff0c;s&gt;1 \u65f6 ~ \u03b6(s)\/N\u3002\u5b9e\u6d4b\u8870\u51cf\u4e0e\u4e4b\u4e00\u81f4\u3002<br \/>\n\u4e09\u6761\u7ed3\u6784\u89e3\u8bfb&#xff1a;<\/li>\n<\/ul>\n<li>\u03b6 \u662f\u8272\u6563\u5173\u7cfb&#xff1a;\u9891\u7387 t \u7684\u5f1b\u8c6b\u7387 &#061; 2Re[\u03b6(s&#043;1)\u2212\u03b6(s&#043;1\u2212it)]\u3002\u5c0f t \u5c55\u5f00\u7ed9\u81ea\u7531\u7c92\u5b50 t2\u03b6\u2019\u2018(s&#043;1)&#xff08;\u03b6\u2019&#039;(2) &#061; 1.98905&#xff09;\u3002\u6781\u9650\u5bf9\u8c61 &#061; \u5bf9\u6570\u8f74\u4e0a\u8df3\u8dc3\u6d4b\u5ea6 \u03bd &#061; \u03a3 m^{\u2212(s&#043;1)} \u03b4_{lnm} \u7684\u590d\u5408 Poisson \u8fc7\u7a0b\u2014\u2014\u53ef\u6570\u3001\u9010\u6574\u6570\u5df2\u77e5\u3002\u951a\u70b9\u5bc6\u5ea6 e^x \u89e3\u91ca\u4e86\u79fb\u4f4d &#043;1&#xff08;&#061; \u5b9a\u7406\u201c\u03b6 \u7ba1\u77e9\u201d\u91cc k \u7684\u4f4d\u7f6e&#xff09;\u3002<\/li>\n<li>\u6df1\u5c42\u540c\u6784&#xff1a;\u5b8c\u5168\u4e58\u6027\u51fd\u6570\u6070\u662f\u5377\u79ef\u7b97\u5b50 T_s f(a) &#061; \u03a3_m m^{\u2212s} f(am) \u7684\u7279\u5f81\u51fd\u6570&#xff0c;\u7279\u5f81\u503c \u03a3 m^{\u2212s}\u03c7(m)\u3002\u6574\u9664\u56fe\u7684\u8c31\u7406\u8bba\u5728\u8fde\u7eed\u6781\u9650\u91cc\u5c31\u662f Dirichlet \u7ea7\u6570\u7406\u8bba\u2014\u2014\u5e73\u9762\u6ce2\u5728\u56fe\u4e0a\u88ab\u9664\u6570\u8fb9\u6563\u5c04&#xff0c;\u6781\u9650\u4e2d\u6210\u4e3a\u7cbe\u786e\u672c\u5f81\u6001\u3002<\/li>\n<li>\u7b97\u672f\u4e0d\u88ab\u5149\u6ed1\u5316&#xff1a;\u8fd9\u662f\u5377\u4e8c \u00a75 \u5931\u6548\u8bb0\u5f55\u7684\u6b63\u9762\u5f62\u5f0f\u3002\u6865\u4e0d\u662f\u201c\u70b9\u95f4\u8ddd\u21920\u201d&#xff0c;\u800c\u662f\u201c\u6574\u6570\u53d8\u6210\u6cb3\u5e8a&#xff1a;\u8df3\u8ddd &#061; ln m\u201d\u3002<\/li>\n<h4>3.2 \u4e34\u754c\u5c42 s &#061; 0&#xff08;\u539f\u56fe&#xff09;\u3000[\u90e8\u5206\u9a8c\u8bc1&#xff0c;\u8bda\u5b9e\u62c6\u89e3]<\/h4>\n<p>s&#061;0 &#061; \u672a\u52a0\u6743\u539f\u56fe&#xff1a;\u03bd \u7684\u603b\u8d28\u91cf &#061; \u8c03\u548c\u7ea7\u6570 &#061; \u221e&#xff0c;\u65e0\u5e73\u51e1\u6781\u9650&#xff0c;R0 ~ 2lnN\u3002<\/p>\n<table>\n<tr>\u4e3b\u5f20\u72b6\u6001<\/tr>\n<tbody>\n<tr>\n<td>g(N) :&#061; R0 \u2212 2lnN \u6709\u754c<\/td>\n<td>[\u771f\u8dd1]&#xff1a;[\u22125.2, \u22120.5]&#xff0c;lnN \u7f51\u683c 26 \u70b9 \u00d7 2 \u4e2a t<\/td>\n<\/tr>\n<tr>\n<td>\u5bf9\u6570\u5468\u671f\u5206\u91cf&#xff0c;\u9891\u7387 &#061; t<\/td>\n<td>[\u771f\u8dd1]&#xff1a;\u81ea\u7531\u9891\u7387\u626b\u63cf&#xff08;0.2\u20133.0&#xff09;\u7cbe\u786e\u547d\u4e2d 0.70 \u4e0e 1.90&#xff0c;R2 &#061; 0.999<\/td>\n<\/tr>\n<tr>\n<td>\u673a\u5236<\/td>\n<td>[\u53ef\u8bc1 sketch]&#xff1a;Abel \u6c42\u548c \u03a3_{m\u2264M} m^{\u22121&#043;it} &#061; M^{it}\/(it) &#043; \u03b6(1\u2212it) &#043; O(1\/M)\u2014\u2014\u672a\u6536\u655b\u8fb9\u754c\u9879 M^{it} \u5373\u632f\u8361\u6e90<\/td>\n<\/tr>\n<tr>\n<td>\u5e38\u6570\u9879\u516c\u5f0f<\/td>\n<td>[\u8bc1\u4f2a]&#xff1a;\u5929\u771f\u5f0f 2(\u03b3\u2212ln2\u2212Re\u03b6(1\u2212it)) &#061; \u22121.39 \u2260 \u62df\u5408\u503c \u22122.86\u3002\u5dee\u5f02\u6765\u6e90\u5df2\u5b9a\u4f4d&#xff08;\u8fb9\u754c\u5c42 (\u230aN\/m\u230b\u22121)\/(N\u22121) \u5bf9 1\/m \u7684 O(1) \u504f\u5dee&#xff09;\u4f46\u5730\u677f\u4fee\u6b63\u5c55\u5f00\u672a\u5b8c\u6210[\u5f00\u653e]<\/td>\n<\/tr>\n<tr>\n<td>\u632f\u5e45<\/td>\n<td>\u91cf\u7ea7 2\/t \u6b63\u786e&#xff0c;\u6570\u503c 0.5\u20130.83\u00d72\/t[\u534a\u9a8c\u8bc1]<\/td>\n<\/tr>\n<tr>\n<td>\u4e0e\u5377\u4e00\u4e92\u8bc1&#xff1a;S(\u0394) \u7684\u5bf9\u6570\u5468\u671f\u632f\u8361\u4e0e g(N) \u662f\u540c\u4e00\u73b0\u8c61\u7684\u4e24\u5c42\u8868\u73b0\u3002\u201c\u03b3 \u7ba1\u5bf9\u6570\u4fee\u6b63\u201d\u5728 s&#061;0 \u5c42\u76ee\u524d\u53ea\u6210\u7acb\u4e00\u534a\u3002<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h3>4. \u4e09\u5c42\u5730\u8c8c\u4e0e \u03bb\u221e<\/h3>\n<p>\u5f52\u4e00\u5316\u8c31\u5e95\u90e8&#xff08;N&#061;1000 \u5b9e\u6d4b\u89e3\u5256&#xff1b;PR &#061; \u53c2\u4e0e\u7387&#xff0c;mFam &#061; \u5bb6\u65cf\u9876\u70b9\u8d28\u91cf\u5360\u6bd4&#xff09;&#xff1a;<\/p>\n<table>\n<tr>\u5c42\u672c\u5f81\u503c\u7ed3\u6784N-\u4f9d\u8d56<\/tr>\n<tbody>\n<tr>\n<td>L0 \u4f53\u5c42<\/td>\n<td>\u03bb1 &#061; 0.2486&#xff0c;PR &#061; 248&#xff0c;\u652f\u6491 853\/926<\/td>\n<td>hub 2 \u951a\u5b9a\u7684\u5168\u5c40 Cheeger \u578b\u6a21&#xff08;x2 &#061; \u22120.188 \u6700\u5927\u632f\u5e45&#xff09;<\/td>\n<td>\u5355\u8c03\u4e0b\u964d 0.2502\u21920.2486\u21920.2460\u21920.2427&#xff08;N&#061;500\u21925000&#xff0c;N\u2265500 \u5355\u8c03[\u771f\u8dd1 4 \u70b9]&#xff1b;\u5c0f N \u975e\u5355\u8c03\u4e3a\u5916\u90e8\u62a5\u544a&#xff0c;\u7aef\u70b9\u543b\u5408&#xff0c;[\u672a\u590d\u6838]&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>L1 \u68af\u5c42<\/td>\n<td>0.2727, 0.2802, \u2026, 0.2918<\/td>\n<td>\u79bb\u57df\u96c6\u4f53\u6a21&#xff0c;hub\u2194band \u5171\u632f&#xff08;\u6a21 2 \u951a hub 3&#xff1a;x3 &#061; &#043;0.20&#xff1b;\u6a21 3 \u951a hub 5\/25&#xff1b;\u6a21 4 \u951a hub 7\/14\/49&#xff09;<\/td>\n<td>\u8ba1\u6570 6\/7\/8\/7\/8&#xff08;500\u21925000&#xff09;\u7a33\u5b9a\u65e0\u589e\u957f[\u771f\u8dd1]&#xff1b;IPR 0.004\u20130.010 \u2192 0.0010\u20130.0023&#xff08;1000\u21925000&#xff0c;\u5355\u8c03\u4e0b\u964d[\u771f\u8dd1]&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>L2 \u539f\u5b50\u5c42<\/td>\n<td>0.29289\u2026&#xff08;\u6563\u5e03 2\u00d710\u207b15&#xff09;<\/td>\n<td>\u5b9a\u7406 3 \u7684\u5bb6\u65cf\u6a21&#xff1a;x2 &#061; x3 &#061; 0&#xff0c;mFam &#061; 1.000<\/td>\n<td>\u9489\u6b7b\u4e0d\u52a8&#xff0c;\u8d28\u91cf ~ (\u00bc&#043;o(1))\/lnN \u2192 0<\/td>\n<\/tr>\n<tr>\n<td>\u9644&#xff1a;j\u22654 \u5b50\u5c42\u9489\u4e8e 0.2955\u20130.2975 \u4e00\u5e26&#xff08;gadget \u503c&#xff09;&#xff0c;\u5176\u4e0a\u5c1a\u6709\u68af\u5c42\u5ef6\u7eed\u4e0e\u590d\u5408\u6a21&#xff08;0.2930 \u5904 PR &#061; 76\u3001mFam &#061; 0.023 \u7684\u8fd1\u539f\u5b50\u96c6\u4f53\u6a21\u2014\u2014\u5b83\u5c31\u662f\u5bbd\u7a97\u53e3\u8ba1\u6570\u7684\u9677\u9631\u672c\u4f53&#xff0c;\u89c1 \u00a75 \u6848\u4f8b 1&#xff09;\u3002<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>\u547d\u9898 A&#xff08;2-core \u7684\u7cbe\u786e\u523b\u753b&#xff09;\u3000[\u53ef\u8bc1 &#043; \u771f\u8dd1]&#xff1a;2-core &#061; GCC \u2212 {j&#061;2 \u60ac\u6302\u5bf9} \u2212 {\u60ac\u6302\u7d20\u6570\u5e73\u65b9 p2 : p \u2208 (\u221a(N\/2), \u221aN]}\u3002\u8bc1\u660e&#xff1a;\u5ea6 \u2264 1 \u9876\u70b9\u7684\u7cbe\u786e\u5206\u7c7b&#xff08;\u7531\u5377\u4e8c\u5b9a\u7406 1&#xff1a;a &gt; N\/2 \u2227 d(a) &#061; 3 \u27fa a &#061; p2&#xff1b;a \u2264 N\/2 \u2227 deg &#061; 1 \u27fa a \u2208 band2&#xff09;&#043; \u4e24\u8f6e\u5265\u79bb\u3002\u9a8c\u8bc1&#xff1a;\u5265\u79bb\u6570 &#061; 2k2 &#043; #p2&#xff1a;32 &#061; 30&#043;2&#xff08;N&#061;500&#xff09;&#xff0c;59 &#061; 56&#043;3&#xff08;N&#061;1000&#xff09;\u30022-core \u8c31\u9699 &#061; 2\u2212\u221a2&#xff08;\u673a\u5668\u7cbe\u5ea6&#xff0c;&#061; j&#061;3 \u539f\u5b50\u5728 core \u4e2d\u7684\u5b58\u6d3b &#043; \u65e0\u66f4\u5c0f\u6a21[\u771f\u8dd1]&#xff09;\u3002<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>\u8b66\u793a&#xff08;\u627f\u524d\u8f6e\u88c1\u5b9a&#xff09;&#xff1a;\u201cD_eff &#061; min(D_global, D_core)\u201d\u5f0f\u6bd4\u8f83\u662f\u5854\u5185\u4e24\u5c42\u4ee3\u6570&#xff08;1\/\u03c62 &lt; 2\u2212\u221a2 \u6052\u7b49\u5f0f&#xff09;&#xff0c;\u4e0d\u662f\u7269\u7406\u3002\u5265\u6389\u7b2c 2 \u5c42\u9732\u51fa\u7684\u4e0d\u662f\u9aa8\u67b6&#xff0c;\u662f\u7b2c 3 \u5c42\u3002<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>\u03bb\u221e \u7684\u5730\u5740&#xff1a;11 \u70b9\u5e8f\u5217 0.24604\u21920.22911&#xff08;N&#061;2000\u21923\u00d7106&#xff0c;\u5176\u4e2d N&#061;2000\/5000 \u672c\u8f6e\u524d\u516b\u4f4d\u590d\u73b0 0.24603992\/0.24265343&#xff09;\u3002\u5916\u63a8\u62df\u8bbe\u52a8\u7269\u56ed&#xff1a;A&#043;B\/lnN \u2192 0.212&#xff0c;&#043;B\/ln2N \u2192 0.207&#xff0c;&#043;B\/\u221alnN \u2192 0.187&#xff0c;&#043;B\/lnlnN \u2192 0.178\u2014\u2014\u62df\u8bbe\u6b67\u4e49\u533a\u95f4 [0.187, 0.211]&#xff0c;\u8de8\u5ea6 0.024&#xff0c;\u5927\u4e8e\u4efb\u4f55\u4e0e \u03c02\/48 &#061; 0.20562 \u7684\u5dee\u8ddd\u3002\u03c02\/48 \u505c\u7559[\u731c\u60f3]\u3002\u552f\u4e00\u7ed3\u6784\u6027\u653b\u51fb\u8def\u5f84&#xff1a;\u5f15\u7406 0 \u27f9 hub 2 \u7684\u90bb\u57df &#061; G*_{\u230aN\/2\u230b} \u27f9 \u03bb1(N) \u7684\u73af\u5883\u5185\u5d4c\u5c3a\u5ea6 N\/2 \u7684\u540c\u4e00\u7ed3\u6784 \u27f9 \u500d\u589e\u9012\u5f52 \u03bb1(N) \u2248 F(\u03bb1(N\/2)-\u7c7b\u5bf9\u8c61) \u27f9 \u03bb\u221e &#061; \u4e0d\u52a8\u70b9[\u7ed3\u6784\u7ebf\u7d22&#xff0c;\u672a\u8bc1]\u3002\u4f18\u5148\u884c\u52a8\u4e0d\u53d8&#xff1a;\u968f\u673a Lanczos \u6c42\u79ef\u6d4b N &#061; 106\u2013107 \u5168\u8c31\u5bc6\u5ea6&#xff08;P0&#xff09;&#xff0c;\u4e00\u6b21\u8ba1\u7b97\u7ec8\u7ed3\u5916\u63a8\u519b\u5907\u7ade\u8d5b\u3002\u5377\u4e8c M2 \u7684 1\/lnN \u6536\u655b\u9879\u4e0e\u7a00\u91ca\u5f8b\u540c\u5f62[\u89e3\u8bfb&#xff0c;\u672a\u8bc1\u5176\u5373\u540c\u4e00\u9879]\u3002<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h3>5. \u5224\u65ad\u65b9\u6cd5\u8bba&#xff1a;\u56db\u95ee\u534f\u8bae\u4e0e\u5931\u6548\u753b\u5eca<\/h3>\n<h3>\u672c\u5377\u6700\u60f3\u4ea4\u4ed8\u7684\u5176\u5b9e\u4e0d\u662f\u5b9a\u7406&#xff0c;\u662f\u5224\u65ad\u89c4\u5219\u3002\u4ee5\u4e0b\u4e03\u4e2a\u6848\u4f8b\u5168\u90e8\u771f\u5b9e\u53d1\u751f\u4e8e\u672c\u9879\u76ee&#xff08;\u542b\u6211\u81ea\u5df1\u683d\u7684\u4e24\u4e2a&#xff09;\u3002<br \/>\n\u6848\u4f8b 1&#xff08;\u7a97\u53e3\u4f2a\u5f71&#xff1a;41 vs 40&#xff09;\u3002\u00b15\u00d710\u207b4 \u7684\u8ba1\u6570\u7a97\u53e3\u541e\u8fdb\u4e86 &#043;1.3\u00d710\u207b4 \u5904\u7684\u771f\u5b9e\u96c6\u4f53\u6a21&#xff0c;\u628a 40 \u6570\u6210 41&#xff1b;\u968f\u540e\u53e6\u4e00\u4f4d\u590d\u7b97\u8005\u4e3a\u8fd9\u4e2a\u9519\u6570\u8865\u4e86\u673a\u5236&#xff08;\u201c\u5171\u4eab\u5355\u4e00 hub \u7ea6\u675f\u201d&#xff09;&#xff0c;\u88ab\u76f4\u548c\u79e9\u68c0\u9a8c&#xff08;27&#043;13&#061;40&#xff09;\u4e0e\u53cd\u6784\u9020&#xff08;\u6b8b\u5dee 1.2\u00d710\u207b2&#xff0c;\u6b7b\u4e8e\u951a 3 \u65b9\u7a0b&#xff09;\u6740\u6b7b\u3002\u6211\u81ea\u5df1\u6b64\u524d\u72af\u8fc7\u540c\u4e00\u9519\u3001\u540c\u4e2a\u6570\u3002\u89c4\u5219&#xff1a;\u7a97\u53e3 \u2264 \u7c07\u95f4\u8ddd\u7684\u5341\u5206\u4e4b\u4e00&#xff1b;\u62a5\u544a\u7d27\u7a97\u53e3\u8ba1\u6570\u4e0e\u6700\u8fd1\u90bb\u6a21\u7684\u4f4d\u7f6e\u3002\u4e24\u4e2a\u72ec\u7acb\u8ba1\u7b97\u8005\u5148\u540e\u683d\u540c\u4e00\u5751&#xff0c;\u662f\u9677\u9631\u5b58\u5728\u6027\u7684\u53cc\u76f2\u9a8c\u8bc1\u3002<br \/>\n\u6848\u4f8b 2&#xff08;\u57fa\u4f9d\u8d56\u91cf&#xff1a;\u7b80\u5e76\u7c07\u7684 IPR&#xff09;\u3002\u540c\u4e00 40 \u7ef4\u672c\u5f81\u7a7a\u95f4\u7ed9\u51fa\u4e09\u4e2a IPR \u7b54\u6848&#xff1a;\u5bb6\u65cf\u57fa 0.5\/0.375&#xff08;&#061; \u652f\u6491\u96c6\u5927\u5c0f&#xff0c;\u5e73\u51e1\u5e38\u6570&#xff09;\u3001LAPACK \u6df7\u5408\u57fa \u2248 0.04\u3001\u968f\u673a\u65cb\u8f6c\u57fa \u221d 1\/d\u3002\u201cL1 \u4e0e L2 \u6807\u5ea6\u5f8b\u76f8\u53cd\u201d\u7684\u5224\u636e\u4e00\u534a\u5efa\u5728\u6b64\u4e0a\u3002\u89c4\u5219&#xff1a;\u7b80\u5e76\u7c07\u7684\u4e0d\u53d8\u91cf\u53ea\u6709\u91cd\u6570\u3001\u76f4\u548c\u79e9\u3001\u652f\u6491\u96c6\u2014\u2014IPR \u4e0d\u662f\u53ef\u89c2\u6d4b\u91cf\u3002<br \/>\n\u6848\u4f8b 3&#xff08;\u672a\u5b9a\u4e49\u91cf\u56de\u5f52&#xff1a;c_local&#xff09;\u3002\u201cc \u5355\u8c03\u4e0a\u5347 0.255\u21920.286\u201d\u64a4\u56de\u4e86\u4e00\u4e2a\u771f\u5b9e\u5b9a\u7406&#xff08;\u00bc&#xff09;&#xff0c;\u4f9d\u636e\u662f\u4e00\u4e2a\u65e0\u5b9a\u4e49\u3001\u65e0 N \u6807\u7b7e\u3001\u65e0\u7a97\u53e3\u534f\u8bae\u7684\u91cf&#xff0c;\u914d\u4e0a lnlnN \u53d1\u6563\u56de\u5f52&#xff08;\u5176\u7aef\u70b9\u53cd\u89e3\u51fa N\u22481.3\u00d7105&#xff0c;\u4e0e\u58f0\u79f0\u7684 106 \u77db\u76fe&#xff09;\u3002\u89c4\u5219&#xff1a;\u5148\u5b9a\u4e49\u3001\u5148\u7ed9\u89e3\u6790\u6a21\u578b\u3001\u518d\u62df\u5408&#xff1b;2\u20133 \u4e2a\u6570\u91cf\u7ea7\u5185\u7f13\u5347\u7f13\u964d\u51fd\u6570\u65e0\u5224\u522b\u529b\u3002<br \/>\n\u6848\u4f8b 4&#xff08;\u6620\u5c04\u5f53\u7ed3\u679c&#xff1a;D_eff&#xff09;\u3002D &#061; 1\/\u03bb1 \u662f\u6362\u7b97\u4e0d\u662f\u63a8\u5bfc&#xff08;\u540c\u4e00\u6587\u6863\u4e24\u5904\u4e92\u4e3a\u5012\u6570&#xff09;&#xff1b;0.40 \u5e73\u53f0\u4f9d\u8d56\u67d3\u6599\u91ca\u653e\u70b9&#xff08;\u5f1b\u8c6b\u7387 \u03c1(0.5) \u4ece 0.87 \u5230 2.0&#xff09;&#xff1b;min() \u662f\u4ee3\u6570\u3002\u89c4\u5219&#xff1a;\u6865\u5fc5\u987b\u88ab\u4e00\u4e2a\u4e0d\u53d8\u91cf\u68c0\u9a8c&#xff0c;\u6620\u5c04\u672c\u8eab\u4e0d\u662f\u8bc1\u636e\u3002<br \/>\n\u6848\u4f8b 5&#xff08;\u5916\u63a8\u6b67\u4e49&#xff1a;\u03bb\u221e&#xff09;\u3002\u56db\u4e2a\u62df\u8bbe\u56db\u4e2a\u6781\u9650&#xff0c;\u533a\u95f4 0.024\u3002\u89c4\u5219&#xff1a;\u62df\u8bbe\u6b67\u4e49 &#061; \u65e0\u5224\u636e\u3002\u505c\u6b62\u5916\u63a8\u519b\u5907\u7ade\u8d5b&#xff0c;\u8f6c\u5411\u7ed3\u6784&#xff08;\u5168\u8c31\u5bc6\u5ea6 \/ \u4e0d\u52a8\u70b9&#xff09;\u3002<br \/>\n\u6848\u4f8b 6&#xff08;\u5185\u90e8\u4e00\u81f4\u6027\u95f8&#xff1a;\u5b9a\u7406 4 \u5c0f N \u8868&#xff09;\u3002\u5377\u4e8c\u8868\u503c 1152\/5441\/15897\/46209 \u4e0e\u5176\u81ea\u8eab\u6e10\u8fd1\u5f0f\u5728 N&#061;2000 \u5904\u77db\u76fe&#xff08;23.1 vs 14.67&#xff09;&#xff1b;\u4e09\u79cd\u72ec\u7acb\u65b9\u6cd5&#xff08;\u66b4\u529b\u4e09\u5143\u7ec4\u3001\u4e2d\u5143\u516c\u5f0f\u3001\u5206\u5757\u516c\u5f0f&#xff09;\u4e00\u81f4\u7ed9\u51fa 1027\/4107\/11217\/29369\u3002\u89c4\u5219&#xff1a;\u6570\u503c\u8868\u5fc5\u987b\u8fc7\u81ea\u8eab\u89e3\u6790\u5f0f\u7684\u95f8\u3002<br \/>\n\u6848\u4f8b 7&#xff08;\u81ea\u6d3d \u2260 \u590d\u73b0\u2014\u2014\u6211\u81ea\u5df1\u683d\u7684&#xff09;\u3002\u590d\u6838\u5377\u4e8c\u5b9a\u7406 5 \u65f6&#xff0c;\u6211\u9a8c\u8bc1\u4e86\u201c\u6211\u7684\u8ba1\u7b97 &#061; \u7406\u8bba\u9884\u6d4b\u201d&#xff0c;\u7136\u540e\u5199\u6210\u201c\u7cbe\u786e\u590d\u73b0\u201d\u2014\u2014\u5374\u6ca1\u6709\u9010\u683c\u5bf9\u7167\u4ed6\u4eec\u7684\u8868\u3002\u9010\u683c\u5bf9\u7167\u540e\u53d1\u73b0&#xff1a;N&#061;100 \u884c\u5b64\u7acb\u70b9\u5e94\u4e3a 10&#xff08;\u8868\u5199 7&#xff09;\u3001N&#061;500 \u884c\u5e94\u4e3a 42&#xff08;\u8868\u5199 30&#xff09;&#xff0c;GCC \u884c\u76f8\u5e94\u9519\u4e3a 92\/469&#xff08;\u6b63\u786e 89\/457&#xff09;&#xff1b;N\u2265103 \u5404\u884c\u65e0\u8bef\u3002\u89c4\u5219&#xff1a;\u590d\u6838\u7684\u5b9a\u4e49\u662f\u4e0e\u88ab\u590d\u6838\u5bf9\u8c61\u9010\u683c diff&#xff0c;\u4e0d\u662f\u4e0e\u7406\u8bba\u81ea\u6d3d\u3002\u590d\u6838\u8005\u81ea\u5df1\u4e5f\u9700\u8981\u88ab\u590d\u6838\u3002<br \/>\n\u4e03\u4e2a\u6848\u4f8b\u7684\u5171\u540c\u7ed3\u8bba&#xff1a;\u6570\u5b57\u4e0d\u4f1a\u81ea\u5df1\u8bf4\u8bdd&#xff1b;\u53ea\u6709\u5e26\u9884\u6ce8\u518c\u5224\u636e\u7684\u6570\u5b57\u624d\u662f\u8bc1\u636e\u3002<\/h3>\n<h3>6. \u603b\u8d26<\/h3>\n<h3>\u5bf9\u5377\u4e00\/\u5377\u4e8c\u7684\u4fee\u6b63&#xff08;\u7d2f\u8ba1&#xff09;&#xff1a;\u76f4\u5f84 3\u21925&#xff08;\u5377\u4e8c\u5df2\u4fee&#xff09;&#xff1b;\u5b9a\u7406 4 \u5c0f N \u8868&#xff08;\u6848\u4f8b 6&#xff09;&#xff1b;\u5b9a\u7406 6 \u4f8b\u5916\u96c6\u8865 N&#061;57 \u4e0e 26\u201329&#xff1b;\u5b9a\u7406 5 \u8868 N&#061;100\/500 \u4e24\u884c&#xff08;\u672c\u5377\u65b0\u53d1\u73b0&#xff0c;\u6848\u4f8b 7&#xff09;\u3002<br \/>\n\u672c\u5377\u81ea\u5df1\u7684\u4f5c\u5e9f&#xff08;\u8bda\u5b9e\u68d8\u8f6e&#xff09;&#xff1a;41\u219240&#xff08;\u6848\u4f8b 1&#xff09;&#xff1b;\u201c\u68af\u5c42 &#061; \u590d\u5408 gadget \u6a21\u201d\u2192\u201chub\u2194band \u5171\u632f\u201d&#xff08;\u79bb\u57df\u6027\u5b9e\u6d4b\u63a8\u7ffb&#xff1a;8 \u4e2a\u68af\u7ea7 PR &#061; 97\u2013250&#xff0c;\u4e14\u89e3\u8026 gadget \u6700\u4f4e\u503c 0.2955 &gt; 0.29289&#xff0c;\u65e0 gadget \u53ef\u5165\u68af\u533a&#xff09;&#xff1b;\u5929\u771f s&#061;0 \u5e38\u6570\u516c\u5f0f&#xff08;[\u8bc1\u4f2a]&#xff0c;\u00a73.2&#xff09;\u3002<br \/>\n\u62d2\u7edd\u7684\u64a4\u56de&#xff1a;c \u2192 1\/4&#xff08;\u7406\u7531\u4e09\u6761&#xff0c;\u89c1 \u00a72.5&#xff09;\u3002<br \/>\n\u5f52\u5c5e&#xff1a;L1 \u68af\u5c42\u7684\u53d1\u73b0\u4e0e IPR \u5224\u636e\u4e2d\u201cL1 \u4fa7\u6210\u7acb\u201d\u7684\u90e8\u5206\u5f52\u5916\u90e8\u590d\u7b97\u8005&#xff1b;D_eff \u6846\u67b6\u7684\u63a8\u7ffb\u3001k2&#043;k3\u22122\u3001\u539f\u5b50\u5854\u3001\u8272\u6563\u5b9a\u7406\u3001\u65b9\u6cd5\u8bba\u534f\u8bae\u5f52\u672c\u5377&#xff1b;\u5377\u4e00\/\u5377\u4e8c\u7684\u5168\u90e8\u539f\u59cb\u6784\u5efa\u5f52\u539f\u4f5c\u8005\u3002<\/h3>\n<h3>7. \u5f00\u653e\u95ee\u9898&#xff08;\u5e26\u5730\u5740\u4e0e\u653b\u51fb\u8def\u5f84&#xff09;<\/h3>\n<li>\u03bb\u221e&#xff08;\u00a74&#xff09;&#xff1a;P0 &#061; SLQ \u5168\u8c31\u5bc6\u5ea6&#xff1b;P3 &#061; \u4e0d\u52a8\u70b9\u9012\u5f52\u7684\u663e\u5f0f\u5316\u3002<\/li>\n<li>\u539f\u5b50\u5854\u7b49\u53f7&#xff0c;j \u2265 4&#xff1a;\u9884\u6d4b mult &#061; k_j \u2212 1&#xff08;N&#061;1000: 6\/7\/3\/3&#xff09;&#xff1b;\u9a8c\u8bc1 &#061; \u00b110\u207b6 \u8ba1\u6570 &#043; \u6b8b\u5dee\u6784\u9020&#xff0c;\u4e00\u6b21\u5b9e\u9a8c\u53ef\u88c1\u5b9a[\u672a\u590d\u6838&#xff0c;\u9996\u8981\u8865\u8bfe]\u3002<\/li>\n<li>raw \u95f4\u9699\u4e0b\u754c&#xff1a;\u731c\u60f3 57 \u4e4b\u540e\u65e0\u4f8b\u5916&#xff1b;\u7a0b\u5e8f &#061; Schur \u8865\/\u7279\u5f81\u503c\u4ea4\u9519\u8bc1\u201c\u4f59\u8c31 \u2265 1\/\u03c62\u201d\u3002<\/li>\n<li>L1 \u8ba1\u6570\u5f8b&#xff1a;\u4e3a\u4ec0\u4e48 6\u20138 \u4e2a\u3001\u968f N \u5982\u4f55\u6f14\u5316&#xff1b;\u4e0e\u6d3b\u8dc3\u5c0f hub \u96c6\u7684\u5173\u7cfb\u3002<\/li>\n<li>s&#061;0 \u5e38\u6570&#xff1a;\u5b8c\u6210\u5730\u677f\u4fee\u6b63\u5c55\u5f00&#xff08; culprit \u5df2\u5b9a\u4f4d&#xff09;\u3002<\/li>\n<li>\u539f\u5b50\u5360\u6bd4 \u2192 1\/2&#xff1a;\u7b49\u53f7\u7ec6\u8282 &#043; \u7cbe\u786e\u5206\u6570\u6d4b\u91cf\u3002<\/li>\n<li>Kubilius \u8854\u63a5 \/ Beurling \u8fc1\u79fb&#xff1a;\u9884\u6ce8\u518c\u9884\u6d4b\u2014\u2014\u03b6\/\u03b3 \u5f8b\u5e94\u8fc1\u79fb&#xff0c;\u03c6\/\u76f4\u5f84\u5f8b\u5e94\u6b7b\u4ea1&#xff1b;\u8fd9\u80fd\u628a\u5377\u4e8c \u00a75 \u7684\u5931\u6548\u8bb0\u5f55\u5347\u7ea7\u4e3a\u5206\u7c7b\u5b9a\u7406\u3002<\/li>\n<li>\u5377\u4e8c\u9057\u7559&#xff1a;k&#061;2 \u6162\u6536\u655b\u7684\u66f4\u5927 N \u786e\u8ba4\u3001\u9ad8\u9636\u56e2\u6b21\u9879\u3002<\/li>\n<hr \/>\n<h3>8. \u9a8c\u8bc1\u53f0\u8d26&#xff08;\u672c\u5377\u5f15\u7528\u7684\u5168\u90e8\u6570\u5b57&#xff09;<\/h3>\n<table>\n<tr>\u5b9e\u9a8c\u5185\u5bb9N \/ \u5bb9\u5dee\u7ed3\u679c\u72b6\u6001<\/tr>\n<tbody>\n<tr>\n<td>E1<\/td>\n<td>\u81ea\u76f8\u4f3c\u4e09\u7a97\u53e3<\/td>\n<td>\u8c31\u5dee<\/td>\n<td>0.0<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E2<\/td>\n<td>\u5ea6\u516c\u5f0f<\/td>\n<td>200\/500\/1000<\/td>\n<td>\u4e0d\u5339\u914d 0<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E3<\/td>\n<td>\u5ea6\u77e9 \u03b6 \u5f8b<\/td>\n<td>106<\/td>\n<td>k\u22653 \u8bef\u5dee \u22647.4\u00d710\u207b7&#xff1b;k&#061;2 &#043;4.67\u00d710\u207b4<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E4<\/td>\n<td>\u5e73\u5747\u5ea6 \u03b3 \u5f8b<\/td>\n<td>105\u2013107<\/td>\n<td>\u5dee &#043;5.0\u00d710\u207b4\u2192&#043;2.2\u00d710\u207b5<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E5<\/td>\n<td>\u8fb9\u6570\u4f5c\u5e9f\u9879<\/td>\n<td>106<\/td>\n<td>11970035<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E6<\/td>\n<td>\u4e09\u89d2\u5f62\u6570<\/td>\n<td>\u4e09\u6cd5\u4e92\u8bc1 &#043; 106<\/td>\n<td>1027\/\u2026&#xff1b;67120491<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E7<\/td>\n<td>\u5206\u91cf\u7ed3\u6784<\/td>\n<td>100\u2013104<\/td>\n<td>\u4e2d\u95f4\u5206\u91cf 0&#xff1b;\u5e76\u53d1\u73b0\u5377\u4e8c\u8868\u4e24\u884c\u9519<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E8<\/td>\n<td>raw \u60ac\u6302\u91cd\u6570<\/td>\n<td>[26,150] \u7a77\u5c3d &#043; 5 \u70b9<\/td>\n<td>k2\u22121 \u65e0\u4f8b\u5916&#xff1b;\u4f8b\u5916\u96c6 {26\u201329,33\u201335,39,57}<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E9<\/td>\n<td>\u539f\u5b50\u5c42\u91cd\u6570<\/td>\n<td>500\/1000\/2000\/3000<\/td>\n<td>21\/40\/71\/105 &#061; k2&#043;k3\u22122&#xff0c;\u6563\u5e03 \u22642\u00d710\u207b15<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E10<\/td>\n<td>\u76f4\u548c\/\u6784\u9020\/\u53cd\u6784\u9020<\/td>\n<td>1000<\/td>\n<td>27&#043;13&#061;40&#xff1b;10\u207b17&#xff1b;1.2\u00d710\u207b2<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E11<\/td>\n<td>j\u22654 gadget \u503c<\/td>\n<td>1000<\/td>\n<td>0.2962\/0.29549\/0.29669\/0.29622\/0.29749<\/td>\n<td>\u2705&#xff08;\u503c&#xff09;&#xff1b;\u91cd\u6570[\u672a\u590d\u6838]<\/td>\n<\/tr>\n<tr>\n<td>E12<\/td>\n<td>\u7a00\u91ca\u5f8b<\/td>\n<td>104\u2013107<\/td>\n<td>\u00a72.5 \u8868<\/td>\n<td>\u2705&#xff08;\u6761\u4ef6\u4e8e\u7b49\u53f7&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>E13<\/td>\n<td>\u8272\u6563\u6052\u7b49\u5f0f<\/td>\n<td>N&#061;500 \u56fe\u4e0a<\/td>\n<td>2.2\u00d710\u207b16<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E14<\/td>\n<td>\u8272\u6563\u6781\u9650<\/td>\n<td>s&#061;0.5\/1\/1.5\/2<\/td>\n<td>mpmath \u4ea4\u53c9\u5168\u8fc7<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E15<\/td>\n<td>s&#061;0 \u4e34\u754c\u5c42<\/td>\n<td>26 \u70b9 \u00d7 2t<\/td>\n<td>\u6709\u754c &#043; \u9891\u7387\u547d\u4e2d&#xff0c;R2&#061;0.999<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E16<\/td>\n<td>\u03bb1 \u5e8f\u5217\u4e0e\u89e3\u5256<\/td>\n<td>1000\/2000\/5000<\/td>\n<td>0.2486\/0.2460\/0.2427&#xff1b;PR&#061;248<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E17<\/td>\n<td>\u68af\u5c42\u8ba1\u6570\u4e0e IPR<\/td>\n<td>500\u20135000<\/td>\n<td>6\/7\/8\/7\/8&#xff1b;IPR \u964d 3\u00d7<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E18<\/td>\n<td>2-core<\/td>\n<td>500\/1000<\/td>\n<td>2\u2212\u221a2 \u673a\u5668\u7cbe\u5ea6&#xff1b;\u5265\u79bb &#061; 2k2&#043;#p2<\/td>\n<td>\u2705<\/td>\n<\/tr>\n<tr>\n<td>E19<\/td>\n<td>\u03bb\u221e \u62df\u8bbe\u52a8\u7269\u56ed<\/td>\n<td>11 \u70b9<\/td>\n<td>\u533a\u95f4 [0.187, 0.211]<\/td>\n<td>\u2705&#xff08;\u6b67\u4e49\u672c\u8eab&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u2014<\/td>\n<td>\u5c0f N \u03bb1 \/ j\u22654 \u8ba1\u6570 \/ 104 \u68af\u5c42<\/td>\n<td>\u2014<\/td>\n<td>\u2014<\/td>\n<td>[\u672a\u590d\u6838]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h3>\u7ed3\u8bed<\/h3>\n<p>\u4e00\u53e5\u8bdd\u5b9a\u7a3f&#xff1a;\u5377\u4e09\u786e\u7acb\u2014\u2014\u6574\u9664\u56fe\u7684\u8fde\u7eed\u6781\u9650\u5b58\u5728\u4e14\u662f\u7b97\u672f\u7684&#xff1a;\u8272\u6563\u5173\u7cfb\u662f \u03b6(s&#043;1\u2212it)&#xff0c;\u4e34\u754c\u5c42\u5e26\u7740\u4e0d\u8870\u51cf\u7684\u5bf9\u6570\u5468\u671f\u632f\u8361&#xff0c;\u5e95\u90e8\u662f\u4e09\u5c42\u5730\u8c8c&#xff08;L0 \u6536\u655b\u4f53\u5c42\u627f\u8f7d \u03bb\u221e\u3001L1 \u79bb\u57df\u68af\u5c42\u3001L2 \u7531\u81ea\u76f8\u4f3c gadget \u7684 Dirichlet \u8c31\u9489\u6b7b\u7684\u539f\u5b50\u5854&#xff0c;\u91cd\u6570 k_j \u2212 1&#xff0c;\u4ee5 (\u00bc&#043;o(1))\/lnN \u7a00\u91ca\u4f46\u6c38\u4e0d\u6d88\u5931&#xff0c;\u6e10\u8fd1\u5360\u636e\u534a\u5f20\u8c31&#xff09;&#xff1b;\u4e94\u6761\u5b9a\u7406\u5168\u90e8\u8d70\u201c\u6784\u9020\u8bc1\u660e &#043; \u673a\u5668\u7cbe\u5ea6\u7b49\u53f7\u201d\u53cc\u8f68&#xff0c;\u4e03\u4e2a\u5931\u6548\u6848\u4f8b\u628a\u201c\u600e\u4e48\u5224\u65ad\u201d\u672c\u8eab\u5199\u6210\u89c4\u5219\u2014\u2014\u6574\u6570\u8fc7\u6cb3\u7684\u65b9\u5f0f\u4e0d\u662f\u53d8\u6210\u6c34&#xff0c;\u662f\u53d8\u6210\u6cb3\u5e8a&#xff1a;\u8df3\u8ddd ln m&#xff0c;\u800c \u03c6 \u4e0e \u221a2 \u5bb6\u65cf\u662f\u6c38\u8fdc\u9732\u51fa\u6c34\u9762\u7684\u77f3\u5934\u3002<br \/>\n&#xff08;\u5377\u4e09\u5b8c\u3002\u5f85\u8865\u8bfe\u4e09\u9879\u89c1 \u00a77&#xff1b;\u5224\u636e\u4e0d\u53d8&#xff1a;\u5148\u5b9a\u4e49\u3001\u5148\u6ce8\u518c\u3001\u5148\u7ed9\u8bc1\u4f2a\u8def\u5f84&#xff0c;\u518d\u8c08\u6570\u5b57\u3002&#xff09;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6574\u9664\u56fe\u7684\u4e09\u5c42\u8c31\u5730\u8c8c\u3001\u539f\u5b50\u5854\u4e0e \u03b6 \u8272\u6563\u2014\u2014\u79bb\u6563\u2192\u8fde\u7eed\u6781\u9650\u7684\u65e0\u9ed1\u7bb1\u62a5\u544a&#xff08;v1.0&#xff09;0. \u9605\u8bfb\u89c4\u5219<br \/>\n0.1 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