{"id":1553,"date":"2025-08-18T20:48:44","date_gmt":"2025-08-18T11:48:44","guid":{"rendered":"https:\/\/daba-no-heya.com\/?p=1553"},"modified":"2025-10-03T21:52:58","modified_gmt":"2025-10-03T12:52:58","slug":"post-1553","status":"publish","type":"post","link":"https:\/\/daba-no-heya.com\/?p=1553","title":{"rendered":"\u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570"},"content":{"rendered":"\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_83 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/daba-no-heya.com\/?p=1553\/#%E5%B0%8E%E5%85%A5\" >\u5c0e\u5165<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/daba-no-heya.com\/?p=1553\/#%E5%AE%9A%E7%BE%A9%E5%9F%9F\" >\u5b9a\u7fa9\u57df<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/daba-no-heya.com\/?p=1553\/#%E7%A9%8D%E5%88%86%E8%A1%A8%E7%A4%BA\" >\u7a4d\u5206\u8868\u793a<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/daba-no-heya.com\/?p=1553\/#%E7%A9%8D%E5%88%86%E8%A1%A8%E7%A4%BA%E3%83%8F%E3%83%B3%E3%82%B1%E3%83%AB%E7%A9%8D%E5%88%86%E8%B7%AF\" >\u7a4d\u5206\u8868\u793a(\u30cf\u30f3\u30b1\u30eb\u7a4d\u5206\u8def)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/daba-no-heya.com\/?p=1553\/#%E8%87%AA%E6%98%8E%E3%81%AA%E9%9B%B6%E7%82%B9\" >\u81ea\u660e\u306a\u96f6\u70b9<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/daba-no-heya.com\/?p=1553\/#%E7%9B%B8%E5%8F%8D%E5%85%AC%E5%BC%8F\" >\u76f8\u53cd\u516c\u5f0f<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%B0%8E%E5%85%A5\"><\/span>\u5c0e\u5165<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b$\\zeta(s)$\u3092\u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570(Riemann zeta function)\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\zeta(s)=\\sum_{n=1}^{\\infty}\\frac{1}{n^s}=1+\\frac{1}{2^s}+\\frac{1}{3^s}+\\cdots\\quad(\\Re(s)&gt;1)<br>$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%AE%9A%E7%BE%A9%E5%9F%9F\"><\/span>\u5b9a\u7fa9\u57df<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u4ee5\u524d\u306e\u8a18\u4e8b\u3067<a href=\"https:\/\/daba-no-heya.com\/?p=1479\" data-type=\"post\" data-id=\"1479\">\u30d0\u30fc\u30bc\u30eb\u554f\u984c<\/a>\u3092\u7d39\u4ecb\u3057\u307e\u3057\u305f\u3002<br>\u30d0\u30fc\u30bc\u30eb\u554f\u984c\u306f$\\zeta(2)$\u306e\u5024\u3092\u6c42\u3081\u308b\u554f\u984c\u3067\u3059\u306d\u3002<br>$s$\u304c2\u3088\u308a\u5927\u304d\u306a\u5b9f\u6570\u306e\u5834\u5408\u306f$\\frac{1}{n^s}$\u304c\u3088\u308a\u6025\u901f\u306b\u5c0f\u3055\u304f\u306a\u3063\u3066\u3044\u304f\u306e\u3067\u3001$\\zeta(s)$\u304c\u53ce\u675f\u3059\u308b\u3053\u3068\u306f\u4e88\u60f3\u3067\u304d\u308b\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$\\zeta(s)$\u304c\u53ce\u675f\u3059\u308b\u304b\u3069\u3046\u304b\u306e\u5883\u754c\u304c\u3069\u3053\u306b\u3042\u308b\u304b\u3068\u3044\u3046\u3068\u3001$s=1$\u306b\u3042\u308a\u307e\u3059\u3002<br>$s$\u3092\u8907\u7d20\u6570\u3068\u3059\u308c\u3070\u3001$\\Re(s)&gt;1$\u306e\u3068\u304d$\\zeta(s)$\u306f\u53ce\u675f\u3057\u307e\u3059\u3002<br>\u307e\u305a\u306f\u3053\u308c\u3092\u8a3c\u660e\u3057\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\">$s=a+bi$\u3068\u304a\u304f\u3068\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>n^{-s}=n^{-a}n^{-ib}=n^{-a}e^{-ib\\log n}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>|n^{-s}|=n^{-a}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$n\\le t\\le n+1$\u306b\u304a\u3044\u3066\u3001$a&gt;1$\u306a\u3089$n^{-a}&gt;0$\u304b\u3064$t^{-a}&gt;0$\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>&amp;n^{-a}\\ge t^{-a}\\ge (n+1)^{-a} \\\\<br>&amp;\\Leftrightarrow \\int_n^{n+1}n^{-a}dt\\ge\\int_n^{n+1}t^{-a}dt\\ge\\int_n^{n+1}(n+1)^{-a}dt \\\\<br>&amp;\\Leftrightarrow n^{-a}\\ge\\int_n^{n+1}t^{-a}dt\\ge(n+1)^{-a}<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$n$\u306b\u3064\u3044\u30661\u304b\u3089$N$\u307e\u3067\u306e\u548c\u3092\u3068\u308b\u3068\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\int_1^{N+1}t^{-a}dt\\ge -1^{-a}+\\sum_{n=1}^{N+1}n^{-a}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\sum_{n=1}^{N+1}n^{-a}&amp;\\le 1+\\int_1^{N+1}t^{-a}dt \\\\<br>&amp;=1+\\frac{1}{a-1}\\left(1-(N+1)^{1-a}\\right)<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$a&gt;1$\u306e\u3068\u304d\u3001$0&lt;(N+1)^{1-a}&lt;1$\u3001\u3064\u307e\u308a$0&lt;1-(N+1)^{1-a}&lt;1$\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\sum_{n=1}^{N+1}n^{-a}&lt;1+\\frac{1}{a-1}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3057\u305f\u304c\u3063\u3066\u3001$a&gt;1$\u306e\u3068\u304d$\\sum_{n=1}^{N+1}n^{-a}$\u306f\u53ce\u675f\u3059\u308b\u306e\u3067\u3001$\\zeta(s)$\u304c$\\Re(s)&gt;1$\u3067\u53ce\u675f\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E7%A9%8D%E5%88%86%E8%A1%A8%E7%A4%BA\"><\/span>\u7a4d\u5206\u8868\u793a<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">$\\zeta(s)$\u306e\u5143\u306e\u5b9a\u7fa9\u5f0f\u3060\u3068$\\Re(s)&gt;1$\u306e\u7bc4\u56f2\u3067\u3057\u304b\u5b9a\u7fa9\u3067\u304d\u307e\u305b\u3093\u304c\u3001\u89e3\u6790\u63a5\u7d9a\u306b\u3088\u3063\u3066\u5b9a\u7fa9\u57df\u3092$s\\ne 1$\u306e\u8907\u7d20\u6570\u306b\u307e\u3067\u5e83\u3052\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br>\u305d\u306e\u524d\u6bb5\u968e\u3068\u3057\u3066\u3001\u307e\u305a\u306f$\\zeta(s)$\u306e\u7a4d\u5206\u8868\u793a\u3092\u5c0e\u51fa\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\">\u30ac\u30f3\u30de\u95a2\u6570$\\Gamma(s)$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\Gamma(s)=\\int_0^\\infty e^{-t}t^{s-1}dt<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3067$t=nx$\u3068\u304a\u304f\u3068\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\Gamma(s)&amp;=\\int_0^\\infty e^{-nx}(nx)^{s-1}\\cdot ndx \\\\<br>&amp;=\\int_0^\\infty n^se^{-nx}x^{s-1}dx \\\\<br>&amp;=n^s\\int_0^\\infty e^{-nx}x^{s-1}dx<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\frac{1}{n^s}=\\frac{1}{\\Gamma(s)}\\int_0^\\infty e^{-nx}x^{s-1}dx<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u308c\u3088\u308a\u3001$\\zeta(s)$\u306f\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\zeta(s)&amp;=\\sum_{n=1}^\\infty\\frac{1}{n^s} \\\\<br>&amp;=\\sum_{n=1}^\\infty\\frac{1}{\\Gamma(s)}\\int_0^\\infty e^{-nx}x^{s-1}dx \\\\<br>&amp;=\\frac{1}{\\Gamma(s)}\\sum_{n=1}^\\infty\\int_0^\\infty e^{-nx}x^{s-1}dx<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\sum_{n=1}^\\infty e^{-nx}x^{s-1}&amp;=x^{s-1}\\sum_{n=1}^\\infty e^{-nx} \\\\<br>&amp;=x^{s-1}\\lim_{n\\to\\infty}\\frac{e^{-x}(1-e^{-xn})}{1-e^{-x}} \\\\<br>&amp;=x^{s-1}\\cdot\\frac{e^{-x}}{1-e^{-x}} \\\\<br>&amp;=\\frac{x^{s-1}}{e^x-1}<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3088\u308a\u3001$\\sum_{n=1}^\\infty e^{-nx}x^{s-1}$\u306f\u4e00\u69d8\u53ce\u675f\u3059\u308b\u306e\u3067\u3001\u7a4d\u5206\u3068\u548c\u306e\u9806\u756a\u3092\u5165\u308c\u66ff\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u3066\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\zeta(s)&amp;=\\frac{1}{\\Gamma(s)}\\int_0^\\infty\\left(\\sum_{n=1}^\\infty e^{-nx}x^{s-1} \\right)dx \\\\<br>&amp;=\\frac{1}{\\Gamma(s)}\\int_0^\\infty\\frac{x^{s-1}}{e^x-1}dx<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E7%A9%8D%E5%88%86%E8%A1%A8%E7%A4%BA%E3%83%8F%E3%83%B3%E3%82%B1%E3%83%AB%E7%A9%8D%E5%88%86%E8%B7%AF\"><\/span>\u7a4d\u5206\u8868\u793a(\u30cf\u30f3\u30b1\u30eb\u7a4d\u5206\u8def)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\int_C\\frac{(-z)^{s-1}}{e^z-1}dz<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3092\u8003\u3048\u307e\u3059\u3002<br>\u7a4d\u5206\u8def$C$\u306f\u3001\u4ee5\u4e0b\u306b\u793a\u3059\u30cf\u30f3\u30b1\u30eb\u7a4d\u5206\u8def(Hankel contour)\u3067\u3059\u3002<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"420\" height=\"414\" src=\"https:\/\/daba-no-heya.com\/wp-content\/uploads\/2025\/08\/hankel_contour.png\" alt=\"\" class=\"wp-image-1558\" srcset=\"https:\/\/daba-no-heya.com\/wp-content\/uploads\/2025\/08\/hankel_contour.png 420w, https:\/\/daba-no-heya.com\/wp-content\/uploads\/2025\/08\/hankel_contour-300x296.png 300w\" sizes=\"auto, (max-width: 420px) 100vw, 420px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">\u5b9f\u8ef8\u306e\u8ca0\u306e\u65b9\u5411\u304b\u3089\u539f\u70b9\u306b\u5411\u304b\u3044\u3001\u534a\u5f84$\\epsilon$\u306e\u5186\u5f27\u3092\u7d4c\u7531\u3057\u3066\u518d\u3073\u5b9f\u8ef8\u306e\u8ca0\u306e\u65b9\u5411\u306b\u623b\u308b\u7a4d\u5206\u8def\u3067\u3059\u3002<br>\u4e0a\u306e\u56f3\u3060\u3068\u5186\u5f27\u306e\u534a\u5f84\u304c\u305d\u308c\u306a\u308a\u306b\u5927\u304d\u3044\u3088\u3046\u306b\u898b\u3048\u307e\u3059\u304c\u3001\u5b9f\u969b\u306b\u306f\u534a\u5f84$\\epsilon$\u306f\u5fae\u5c0f\u306a\u5024\u3092\u3068\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u307e\u305a\u306f$C_1$\u306e\u7a4d\u5206\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<br>$-z=re^{-i\\pi}$\u3068\u304a\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\int_{C_1}\\frac{(-z)^{s-1}}{e^z-1}dz&amp;=\\int_\\infty^0\\frac{(re^{-i\\pi})^{s-1}}{e^{-re^{-i\\pi}}-1}\\cdot (-e^{-i\\pi})dr \\\\<br>&amp;=\\int_\\infty^0\\frac{r^{s-1}e^{-i\\pi s}e^{i\\pi}}{e^r-1}\\cdot (-e^{-i\\pi})dr \\\\<br>&amp;=-\\int_\\infty^0\\frac{r^{s-1}e^{-i\\pi s}}{e^r-1}dr \\\\<br>&amp;=-e^{-i\\pi s}\\int_\\infty^0\\frac{r^{s-1}}{e^r-1}dr \\\\<br>&amp;=e^{-i\\pi s}\\int_0^\\infty\\frac{r^{s-1}}{e^r-1} \\\\<br>&amp;=e^{-i\\pi s}\\Gamma(s)\\zeta(s)<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u540c\u69d8\u306b\u3057\u3066$C_3$\u306e\u7a4d\u5206\u3082\u8a08\u7b97\u3057\u307e\u3059\u3002<br>$-z=re^{i\\pi}$\u3068\u304a\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\int_{C_3}\\frac{(-z)^{s-1}}{e^z-1}dz&amp;=\\int_0^\\infty\\frac{(re^{i\\pi})^{s-1}}{e^{-re^{i\\pi}}-1}\\cdot (-e^{i\\pi})dr \\\\<br>&amp;=-\\int_0^\\infty\\frac{r^{s-1}e^{i\\pi s}e^{-i\\pi}}{e^r-1}\\cdot e^{i\\pi}dr \\\\<br>&amp;=-e^{i\\pi s}\\int_0^\\infty\\frac{r^{s-1}}{e^r-1}dr \\\\<br>&amp;=-e^{i\\pi s}\\Gamma(s)\\zeta(s)<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6700\u5f8c\u306b$C_2$\u306e\u7a4d\u5206\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<br>$-z=\\epsilon e^{i\\theta}$\u3068\u304a\u3044\u3066\u3001$\\theta$\u3092$-\\pi$\u304b\u3089$\\pi$\u3078\u5909\u5316\u3055\u305b\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\left|\\int_{C_2}\\frac{(-z)^{s-1}}{e^z-1}dz\\right|&amp;=\\left|\\int_{-\\pi}^\\pi\\frac{(\\epsilon e^{i\\theta})^{s-1}}{e^{-\\epsilon e^{i\\theta}}-1}\\cdot (-i\\epsilon e^{i\\theta})d\\theta\\right| \\\\<br>&amp;=\\left|\\int_{-\\pi}^{\\pi}\\frac{\\epsilon^{s-1}e^{i\\theta s}e^{-i\\theta}}{e^{-\\epsilon e^{i\\theta}}-1}\\cdot (-i\\epsilon e^{i\\theta})d\\theta\\right| \\\\<br>&amp;=\\left|-i\\epsilon^s\\int_{-\\pi}^{\\pi}\\frac{e^{i\\theta s}}{e^{-\\epsilon e^{i\\theta}}-1}d\\theta\\right| \\\\<br>&amp;=\\epsilon^{\\Re(s)}\\left|\\int_{-\\pi}^\\pi\\frac{e^{i\\theta s}}{e^{-\\epsilon e^{i\\theta}}-1}d\\theta\\right| \\\\<br>&amp;\\le\\epsilon^{\\Re(s)}\\int_{-\\pi}^\\pi\\left|e^{i\\theta s}\\right|\\left|\\frac{1}{e^{-\\epsilon e^{i\\theta}}-1}\\right|d\\theta \\\\<br>&amp;=\\epsilon^{\\Re(s)}\\int_{-\\pi}^\\pi\\left|e^{i\\theta s}\\right|\\left|\\frac{1}{1-e^{-\\epsilon e^{i\\theta}}}\\right|d\\theta \\\\<br>&amp;=\\epsilon^{\\Re(s)-1}\\int_{-\\pi}^\\pi\\left|e^{i\\theta s}\\right|\\left|\\frac{-\\epsilon e^{i\\theta}}{1-e^{-\\epsilon e^{i\\theta}}}\\right|d\\theta<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30ed\u30d4\u30bf\u30eb\u306e\u5b9a\u7406\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\lim_{z\\to 0}\\left|\\frac{z}{1-e^z}\\right|=\\lim_{z\\to 0}\\left|\\frac{1}{-e^z}\\right|=1<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\lim_{\\epsilon\\to 0}\\left|\\frac{-\\epsilon e^{i\\theta}}{1-e^{-\\epsilon e^{i\\theta}}}\\right|=1<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\int_{-\\pi}^\\pi\\left|e^{i\\theta s}\\right|\\left|\\frac{-\\epsilon e^{i\\theta}}{1-e^{-\\epsilon e^{i\\theta}}}\\right|d\\theta<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u306f\u6709\u754c\u3067\u3001\u3053\u306e\u5024\u3092$M$\u3068\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\left|\\int_{C_2}\\frac{(-z)^{s-1}}{e^z-1}dz\\right|\\le\\epsilon^{\\Re(s)-1}M\\to0\\quad(\\epsilon\\to0,\\Re(s)&gt;1)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4ee5\u4e0a\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\int_C\\frac{(-z)^{s-1}}{e^z-1}dz&amp;=e^{-i\\pi s}\\Gamma(s)\\zeta(s)-e^{i\\pi s}\\Gamma(s)\\zeta(s) \\\\<br>&amp;=\\left(e^{-i\\pi s}-e^{i\\pi s}\\right)\\Gamma(s)\\zeta(s) \\\\<br>&amp;=-2i\\sin(\\pi s)\\Gamma(s)\\zeta(s)<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$\\Gamma(s)\\Gamma(1-s)=\\frac{\\pi}{\\sin(\\pi s)}$\u3088\u308a$\\Gamma(s)\\sin(\\pi s)=\\frac{\\pi}{\\Gamma(1-s)}$\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\int_C\\frac{(-z)^{s-1}}{e^z-1}dz=-\\frac{2\\pi i}{\\Gamma(1-s)}\\zeta(s)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\zeta(s)=-\\frac{\\Gamma(1-s)}{2\\pi i}\\int_C\\frac{(-z)^{s-1}}{e^z-1}dz\\quad(\\Re(s)&gt;1)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u53f3\u8fba\u306e\u7a4d\u5206\u306b\u3064\u3044\u3066\u306f\u3001\u7a4d\u5206\u8def$C$\u306f\u88ab\u7a4d\u5206\u95a2\u6570\u306e\u6975\u3092\u901a\u3089\u306a\u3044\u306e\u3067\u3001$s$\u306e\u5024\u306b\u3088\u3089\u305a\u8907\u7d20\u5e73\u9762\u5168\u4f53\u3067\u6b63\u5247\u3067\u3059\u3002<br>\u4e00\u65b9\u3001$\\Gamma(1-s)$\u306f$s=1,2,3,\\cdots$\u3067\u6975\u3092\u3082\u3064\u306e\u3067\u3001\u3053\u308c\u3092\u9664\u3044\u305f\u9818\u57df$D=\\mathbb{C}\\setminus\\mathbb{N}\\space(s\\ne1,2,3,\\cdots)$\u3092\u8003\u3048\u307e\u3059\u3002<br>\u3053\u306e\u3068\u304d\u3001\u5143\u306e\u5b9a\u7fa9\u57df\u3092\u958b\u96c6\u5408$O=\\{s\\in\\mathbb{C}\\mid\\Re(s)&gt;1,s\\notin\\mathbb{N}\\}$\u3068\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$O\\subset D$<\/li>\n\n\n\n<li>$\\zeta(s)$\u306f$D$\u3067\u6b63\u5247<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">\u306a\u306e\u3067\u3001\u4e00\u81f4\u306e\u5b9a\u7406\u3088\u308a\u3001$\\zeta(s)$\u306f\u9818\u57df$D$\u3067\u3082\u6210\u7acb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\zeta(s)=<br>\\left\\{<br>\\begin{aligned}<br>&amp;\\sum_{n=1}^\\infty\\frac{1}{n^s}\\quad(\\Re(s)&gt;1) \\\\<br>&amp;-\\frac{\\Gamma(1-s)}{2\\pi i}\\int_C\\frac{(-z)^{s-1}}{e^z-1}dz\\quad(s\\ne 1,2,3,\\cdots)<br>\\end{aligned}<br>\\right.<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u306e\u4e8c\u3064\u306e\u5f0f\u3092\u5408\u308f\u305b\u308b\u3068\u3001$\\zeta(s)$\u3092$s\\ne 1$\u306e\u3059\u3079\u3066\u306e\u8907\u7d20\u6570$s$\u306b\u3064\u3044\u3066\u5b9a\u7fa9\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br>\u3053\u306e\u5f8c\u306e\u8a08\u7b97\u306e\u90fd\u5408\u3067$s$\u306b\u5236\u9650\u3092\u304b\u3051\u308b\u3053\u3068\u304c\u3042\u308a\u307e\u3059\u304c\u3001$\\zeta(s)$\u81ea\u4f53\u306f$s\\ne 1$\u306b\u3064\u3044\u3066\u5b9a\u7fa9\u3067\u304d\u308b\u3053\u3068\u3092\u982d\u306e\u7247\u9685\u306b\u7f6e\u3044\u3066\u304a\u3044\u3066\u3082\u3089\u3048\u308b\u3068\u3044\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E8%87%AA%E6%98%8E%E3%81%AA%E9%9B%B6%E7%82%B9\"><\/span>\u81ea\u660e\u306a\u96f6\u70b9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">$n$\u3092\u81ea\u7136\u6570\u3068\u3057\u3066$s=-n$\u3068\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\zeta(-n)&amp;=-\\frac{\\Gamma(1+n)}{2\\pi i}\\oint_C\\frac{(-z)^{-n-1}}{e^z-1}dz \\\\<br>&amp;=-\\frac{n!}{2\\pi i}\\oint_C\\frac{(-z)^{-n-2}}{e^z-1}\\cdot (-z)dz \\\\<br>&amp;=\\frac{in!}{2\\pi}\\oint_C\\frac{-z}{e^z-1}\\cdot\\frac{1}{(-z)^{n+2}}dz \\\\<br>&amp;=\\frac{in!}{2\\pi}\\oint_C\\frac{-1}{(-1)^{n+2}z^{n+2}}\\cdot\\frac{z}{e^z-1}dz \\\\<br>&amp;=\\frac{in!}{2\\pi}(-1)^{n+1}\\oint_C\\frac{1}{z^{n+2}}\\cdot\\frac{z}{e^z-1}dz<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7559\u6570\u5b9a\u7406\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>&amp;\\quad\\frac{in!}{2\\pi}(-1)^{n+1}\\oint_C\\frac{1}{z^{n+2}}\\cdot\\frac{z}{e^z-1}dz \\\\<br>&amp;=\\frac{in!}{2\\pi}(-1)^{n+1}\\cdot 2\\pi i\\mathrm{Res}\\left(\\frac{1}{z^{n+2}}\\cdot\\frac{z}{e^z-1},0\\right) \\\\<br>&amp;=-n!(-1)^{n+1}\\mathrm{Res}\\left(\\frac{1}{z^{n+2}}\\cdot\\frac{z}{e^z-1},0\\right) \\\\<br>&amp;=n!(-1)^n\\mathrm{Res}\\left(\\frac{1}{z^{n+2}}\\cdot\\frac{z}{e^z-1},0\\right)<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30d9\u30eb\u30cc\u30fc\u30a4\u6570\u306e\u5b9a\u7fa9\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>&amp;\\quad n!(-1)^n\\mathrm{Res}\\left(\\frac{1}{z^{n+2}}\\cdot\\frac{z}{e^z-1},0\\right) \\\\<br>&amp;=n!(-1)^n\\mathrm{Res}\\left(\\frac{1}{z^{n+2}}\\sum_{k=0}^{\\infty}\\frac{B_k}{k!}z^k,0\\right) \\\\<br>&amp;=n!(-1)^n\\mathrm{Res}\\left(\\sum_{k=0}^{\\infty}\\frac{B_k}{k!}z^{k-n-2},0\\right)<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30ed\u30fc\u30e9\u30f3\u5c55\u958b\u3057\u305f\u3068\u304d\u306e-1\u6b21\u306e\u4fc2\u6570\u304c\u7559\u6570\u306a\u306e\u3067\u3001$k-n-2=-1$\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\mathrm{Res}\\left(\\sum_{k=0}^{\\infty}\\frac{B_k}{k!}z^{k-n-2},0\\right)=\\frac{B_{n+1}}{(n+1)!}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\zeta(-n)=n!(-1)^n\\cdot\\frac{B_{n+1}}{(n+1)!}=\\frac{(-1)^n B_{n+1}}{n+1}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$B_{n+1}$\u306f$n$\u304c\u5076\u6570\u306e\u3068\u304d\u306b0\u306b\u306a\u308b\u306e\u3067\u3001$\\zeta(s)$\u306f$s$\u304c\u8ca0\u306e\u5076\u6570\u306e\u3068\u304d\u306b0\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<br>\u3053\u308c\u3092\u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570\u306e<strong>\u81ea\u660e\u306a\u96f6\u70b9<\/strong>(trivial zero)\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E7%9B%B8%E5%8F%8D%E5%85%AC%E5%BC%8F\"><\/span>\u76f8\u53cd\u516c\u5f0f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">$\\zeta(s)$\u3068$\\zeta(1-s)$\u306e\u95a2\u4fc2\u3092\u8868\u3059\u76f8\u53cd\u516c\u5f0f(reflection formula)\u3092\u5c0e\u51fa\u3057\u307e\u3059\u3002<br>\u76f8\u53cd\u516c\u5f0f\u306e\u5c0e\u51fa\u306e\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u7a4d\u5206\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\int_{C_K}\\frac{(-z)^{s-1}}{e^z-1}dz<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7a4d\u5206\u8def$C_K$\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u3082\u306e\u3067\u3001\u9375\u7a74\u7a4d\u5206\u8def(keyhole contour)\u3068\u547c\u3070\u308c\u308b\u3053\u3068\u304c\u3042\u308a\u307e\u3059\u3002<br>\u30cf\u30f3\u30b1\u30eb\u7a4d\u5206\u8def\u306e\u5916\u5074\u306b\u3001\u534a\u5f84$R$\u306e\u5186\u5f27\u3092\u8ffd\u52a0\u3057\u305f\u7a4d\u5206\u8def\u3067\u3059\u3002<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"420\" height=\"414\" src=\"https:\/\/daba-no-heya.com\/wp-content\/uploads\/2025\/08\/keyhole_contour.png\" alt=\"\" class=\"wp-image-1576\" srcset=\"https:\/\/daba-no-heya.com\/wp-content\/uploads\/2025\/08\/keyhole_contour.png 420w, https:\/\/daba-no-heya.com\/wp-content\/uploads\/2025\/08\/keyhole_contour-300x296.png 300w\" sizes=\"auto, (max-width: 420px) 100vw, 420px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">$s$\u3092$s&gt;1$\u306e\u5b9f\u6570\u3068\u3057\u307e\u3059\u3002<br>\u3053\u306e\u3068\u304d\u3001$\\frac{(-z)^{s-1}}{e^z-1}$\u306f$z=\\pm 2n\\pi i$ ($n$\u306f\u81ea\u7136\u6570)\u3067\u6975\u3092\u3082\u3061\u307e\u3059\u3002<br>$C_R$\u306e\u534a\u5f84$R$\u3092$R=(2N+1)\\pi$ ($N$\u306f\u81ea\u7136\u6570)\u3068\u3059\u308b\u3053\u3068\u3067\u3053\u306e\u6975\u3092\u901a\u3089\u306a\u3044\u3088\u3046\u306b\u3067\u304d\u308b\u306e\u3067\u3001\u4eca\u56de\u306f\u3053\u306e\u5024\u3092\u63a1\u7528\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u3053\u3067\u306e\u8a08\u7b97\u306e\u65b9\u91dd\u3067\u3059\u304c\u3001\u7559\u6570\u5b9a\u7406\u3092\u7528\u3044\u305f\u5024\u3068\u76f4\u63a5\u7a4d\u5206\u3092\u8a08\u7b97\u3057\u305f\u5024\u3092\u6bd4\u8f03\u3059\u308b\u3053\u3068\u3067\u3001\u76f8\u53cd\u516c\u5f0f\u3092\u5c0e\u51fa\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\">\u307e\u305a\u306f\u7559\u6570\u5b9a\u7406\u3092\u7528\u3044\u305f\u8a08\u7b97\u3067\u3059\u3002<br>\u7559\u6570\u3092\u6c42\u3081\u308b\u969b\u306b\u6975\u306e\u4f4d\u6570\u304c\u5fc5\u8981\u306a\u306e\u3067\u3001\u307e\u305a\u306f\u305d\u308c\u3092\u6c42\u3081\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3042\u308b\u95a2\u6570$f(z)$\u304c$z=a$\u3067$n$\u4f4d\u306e\u96f6\u70b9\u3092\u3082\u3064\u3068\u304d\u3001$z=a$\u306f$\\frac{1}{f(z)}$\u306e$n$\u4f4d\u306e\u6975\u3068\u306a\u308a\u307e\u3059\u3002<br>$f(z)$\u304c$z=a$\u3067$n$\u4f4d\u306e\u96f6\u70b9\u3092\u3082\u3064\u3068\u3044\u3046\u306e\u306f\u3001$f(z)$\u3092\u30ed\u30fc\u30e9\u30f3\u5c55\u958b\u3057\u305f\u3068\u304d\u306b<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>f(z)&amp;=\\sum_{k=n}^{\\infty}a_k(z-a)^k \\\\<br>&amp;=a_n(z-a)^n+a_{n+1}(z-a)^{n+1}+a_{n+2}(z-a)^{n+2}+\\cdots<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u8868\u3055\u308c\u308b\u3068\u3044\u3046\u3053\u3068\u306a\u306e\u3067\u3001$f^{(n)}(a)\\ne0$\u3068\u306a\u308a\u307e\u3059\u3002<br>($a_n(z-a)^n$\u306e\u9805\u306e\u5fae\u5206\u304c0\u3067\u306f\u306a\u304f\u306a\u308b\u305f\u3081)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4eca\u56de\u306e\u5834\u5408\u3001$f(z)=e^z-1$\u3068\u304a\u304f\u3068\u3001$f'(z)=e^z$\u3067\u3059\u3002<br>$f'(\\pm 2n\\pi i)=1\\ne 0$\u3088\u308a\u3001\u6975\u306e\u4f4d\u6570\u306f1\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7559\u6570\u5b9a\u7406\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>-\\frac{1}{2\\pi i}\\int_{C_R}\\frac{(-z)^{s-1}}{e^z-1}dz=\\sum_{n=1}^N\\left(\\mathrm{Res}\\left(\\frac{(-z)^{s-1}}{e^z-1},2n\\pi i\\right)+\\mathrm{Res}\\left(\\frac{(-z)^{s-1}}{e^z-1},-2n\\pi i\\right)\\right)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5de6\u8fba\u306e\u7b26\u53f7\u304c\u30de\u30a4\u30ca\u30b9\u306b\u306a\u3063\u3066\u3044\u308b\u306e\u306f\u3001$C_R$\u306e\u7a4d\u5206\u306e\u5411\u304d\u304c$C_2$\u306e\u9006\u3060\u304b\u3089\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\mathrm{Res}\\left(\\frac{(-z)^{s-1}}{e^z-1},2n\\pi i\\right)&amp;=\\lim_{z\\to 2n\\pi i}(z-2n\\pi i)\\frac{(-z)^{s-1}}{e^z-1} \\\\<br>&amp;=\\lim_{z\\to 2n\\pi i}\\frac{(-z)^{s-1}+(z-2n\\pi i)(s-1)(-z)^{s-2}(-1)}{e^z} \\\\<br>&amp;=(-2n\\pi i)^{s-1} \\\\<br>&amp;=(2n\\pi)^{s-1}(-i)^{s-1} \\\\<br>&amp;=\\frac{(2n\\pi)^{s-1}}{-i}(-i)^s \\\\<br>&amp;=i(2n\\pi)^{s-1}e^{-i\\frac{\\pi}{2}s}<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u540c\u69d8\u306e\u8a08\u7b97\u3067\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\mathrm{Res}\\left(\\frac{(-z)^{s-1}}{e^z-1},-2n\\pi i\\right)&amp;=\\lim_{z\\to 2n\\pi i}(z+2n\\pi i)\\frac{(-z)^{s-1}}{e^z-1} \\\\<br>&amp;=(2n\\pi i)^{s-1} \\\\<br>&amp;=(2n\\pi)^{s-1}i^{s-1} \\\\<br>&amp;=\\frac{(2n\\pi)^{s-1}}{i}i^s \\\\<br>&amp;=-i(2n\\pi)^{s-1}e^{i\\frac{\\pi}{2}s}<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u306a\u308b\u306e\u3067\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>&amp;\\quad\\mathrm{Res}\\left(\\frac{(-z)^{s-1}}{e^z-1},2n\\pi i\\right)+\\mathrm{Res}\\left(\\frac{(-z)^{s-1}}{e^z-1},-2n\\pi i\\right) \\\\<br>&amp;=i(2n\\pi)^{s-1}e^{-i\\frac{\\pi}{2}s}-i(2n\\pi)^{s-1}e^{i\\frac{\\pi}{2}s} \\\\<br>&amp;=i(2n\\pi)^{s-1}\\left(e^{-i\\frac{\\pi}{2}s}-e^{i\\frac{\\pi}{2}s}\\right) \\\\<br>&amp;=i(2n\\pi)^{s-1}\\cdot\\left(-2i\\sin(\\frac{\\pi s}{2})\\right) \\\\<br>&amp;=2\\cdot (2n\\pi)^{s-1}\\sin(\\frac{\\pi s}{2}) \\\\<br>&amp;=2^s (n\\pi)^{s-1}\\sin(\\frac{\\pi s}2{})<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>-\\frac{1}{2\\pi i}\\int_{C_R}\\frac{(-z)^{s-1}}{e^z-1}dz&amp;=\\sum_{n=1}^N\\left(2^s(n\\pi)^{s-1}\\sin(\\frac{\\pi s}{2})\\right) \\\\<br>&amp;=2^s\\pi^{s-1}\\sin(\\frac{\\pi s}{2})\\sum_{n=1}^N n^{s-1} \\\\<br>&amp;=2^s\\pi^{s-1}\\sin(\\frac{\\pi s}{2})\\sum_{n=1}^N\\frac{1}{n^{1-s}}<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$N\\to\\infty$\u3068\u3059\u308c\u3070\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>-\\frac{1}{2\\pi i}\\int_{C_R}\\frac{(-z)^{s-1}}{e^z-1}dz=2^s\\pi^{s-1}\\sin(\\frac{\\pi s}{2})\\zeta(1-s)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$C_2$\u306b\u3064\u3044\u3066\u3082\u7559\u6570\u5b9a\u7406\u3092\u7528\u3044\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\frac{1}{2\\pi i}\\int_{C_2}\\frac{(-z)^{s-1}}{e^z-1}dz&amp;=\\mathrm{Res}\\left(\\frac{(-z)^{s-1}}{e^z-1},0\\right) \\\\<br>&amp;=\\lim_{z\\to 0}z\\cdot\\frac{(-z)^{s-1}}{e^z-1} \\\\<br>&amp;=\\lim_{z\\to 0}\\frac{(-1)^{s-1}z^s}{e^z-1} \\\\<br>&amp;=\\lim_{z\\to 0}\\frac{(-1)^{s-1}sz^{s-1}}{e^z} \\\\<br>&amp;=0<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3057\u305f\u304c\u3063\u3066\u3001$C_K$\u5168\u4f53\u3067\u306f\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\frac{1}{2\\pi i}\\int_{C_K}&amp;=\\frac{1}{2\\pi i}\\int_{C_R}+\\frac{1}{2\\pi i}\\int_{C_2} \\\\<br>&amp;=-2^s\\pi^{s-1}\\sin(\\frac{\\pi s}{2})\\zeta(1-s)<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\">\u7559\u6570\u5b9a\u7406\u3092\u4f7f\u308f\u305a\u306b$\\int_{C_K}$\u3092\u6c42\u3081\u307e\u3059\u3002<br>$C_R$\u4ee5\u5916\u306f\u3059\u3067\u306b\u6c42\u3081\u3066\u3042\u308b\u306e\u3067\u3001$\\int_{C_K}=\\int_C+\\int_{C_R}$\u3067\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$-z=Re^{i\\theta}$\u3068\u304a\u3044\u3066\u3001$\\theta$\u3092$\\pi$\u304b\u3089$-\\pi$\u3078\u5909\u5316\u3055\u305b\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\left|\\int_{C_R}\\frac{(-z)^{s-1}}{e^z-1}dz\\right|&amp;=\\left|\\int_\\pi^{-\\pi}\\frac{\\left(Re^{i\\theta}\\right)^{s-1}}{e^{-iRe^{i\\theta}}-1}\\cdot(-iRe^{i\\theta})d\\theta\\right| \\\\<br>&amp;=\\left|\\int_\\pi^{-\\pi}\\frac{R^{s-1}e^{i\\theta s}e^{-i\\theta}}{e^{-iRe^{i\\theta}}-1}\\cdot(-iRe^{i\\theta})d\\theta\\right| \\\\<br>&amp;=\\left|-iR^s\\int_\\pi^{-\\pi}\\frac{e^{i\\theta s}}{e^{-iRe^{i\\theta}}-1}d\\theta\\right| \\\\<br>&amp;=R^{\\Re(s)}\\left|\\int_\\pi^{-\\pi}\\frac{e^{i\\theta s}}{e^{-iRe^{i\\theta}}-1}d\\theta\\right| \\\\<br>&amp;\\le R^{\\Re(s)}\\int_\\pi^{-\\pi}\\left|e^{i\\theta s}\\right|\\left|\\frac{1}{e^{-iRe^{i\\theta}}-1}\\right|d\\theta<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\lim_{R\\to\\infty}\\left|\\frac{1}{e^{-iRe^{i\\theta}}-1}\\right|=1<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\int_\\pi^{-\\pi}\\left|e^{i\\theta s}\\right|\\left|\\frac{1}{e^{-iRe^{i\\theta}}-1}\\right|d\\theta<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u306f\u6709\u754c\u3067\u3001\u3053\u306e\u5024\u3092$M$\u3068\u304a\u304f\u3068\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\left|\\int_{C_R}\\frac{(-z)^{s-1}}{e^z-1}dz\\right|\\le R^{\\Re(s)}M\\to 0\\quad(R\\to\\infty,\\Re(s)&lt;0)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\int_{C_K}=\\int_C+\\int_{C_R}=\\int_C=-\\frac{2\\pi i}{\\Gamma(1-s)}\\zeta(s)<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\">\u7559\u6570\u5b9a\u7406\u3092\u7528\u3044\u3066\u8a08\u7b97\u3057\u305f\u5024\u3068\u76f4\u63a5\u8a08\u7b97\u3057\u305f\u5024\u3092\u6bd4\u8f03\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7559\u6570\u5b9a\u7406\u3092\u7528\u3044\u3066\u8a08\u7b97\u3057\u305f\u5024<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\frac{1}{2\\pi i}\\int_{C_K}\\frac{(-z)^{s-1}}{e^z-1}dz=-2^s\\pi^{s-1}\\sin(\\frac{\\pi s}{2})\\zeta(1-s)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u76f4\u63a5\u8a08\u7b97\u3057\u305f\u5024<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\frac{1}{2\\pi i}\\int_{C_K}\\frac{(-z)^{s-1}}{e^z-1}dz=\\frac{1}{2\\pi i}\\left(-\\frac{2\\pi i}{\\Gamma(1-s)}\\zeta(s)\\right)=-\\frac{\\zeta(s)}{\\Gamma(1-s)}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>-2^s\\pi^{s-1}\\sin(\\frac{\\pi s }{2})\\zeta(1-s)=-\\frac{\\zeta(s)}{\\Gamma(1-s)}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\zeta(s)=2^s\\pi^{s-1}\\sin(\\frac{\\pi s}{2})\\Gamma(1-s)\\zeta(1-s)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$s=-2n$ ($n$\u306f\u81ea\u7136\u6570)\u3092\u3053\u306e\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\zeta(-2n)=2^{-2n}\\pi^{-2n-1}\\sin(-\\pi n)\\Gamma(1+2n)\\zeta(1+2n)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3082\u3001$s$\u304c\u8ca0\u306e\u5076\u6570\u306e\u3068\u304d\u306b$\\zeta(s)=0$\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5c0e\u5165 \u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b$\\zeta(s)$\u3092\u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570(Riemann zeta function)\u3068\u547c\u3073\u307e\u3059\u3002 $$\\zeta(s)=\\sum_{n=1}^{\\infty}\\frac{1}{n^s}=1+ [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[30],"tags":[],"class_list":["post-1553","post","type-post","status-publish","format-standard","hentry","category-30"],"_links":{"self":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/1553","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1553"}],"version-history":[{"count":30,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/1553\/revisions"}],"predecessor-version":[{"id":1622,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/1553\/revisions\/1622"}],"wp:attachment":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1553"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1553"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1553"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}