{"id":65898,"date":"2025-12-21T22:39:58","date_gmt":"2025-12-21T13:39:58","guid":{"rendered":"https:\/\/wordpress.realcollector.jp\/blog\/?p=65898"},"modified":"2025-12-21T22:39:58","modified_gmt":"2025-12-21T13:39:58","slug":"zeta-3","status":"publish","type":"post","link":"https:\/\/wordpress.realcollector.jp\/blog\/?p=65898","title":{"rendered":"zeta"},"content":{"rendered":"<p>zeta\uff08\u30bc\u30fc\u30bf\uff09\u3068\u306f\u3001\u4e3b\u306b\u6570\u5b66\u306e\u89e3\u6790\u7684\u624b\u6cd5\u3084\u6570\u8ad6\u306b\u304a\u3044\u3066\u767b\u5834\u3059\u308b\u91cd\u8981\u306a\u95a2\u6570\u3067\u3042\u308b\u300c\u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570\uff08Riemann Zeta Function\uff09\u300d\u3092\u6307\u3057\u307e\u3059\u3002\u30d9\u30eb\u30f3\u30cf\u30eb\u30c8\u30fb\u30ea\u30fc\u30de\u30f3\u306b\u3088\u3063\u30661859\u5e74\u306b\u5c0e\u5165\u3055\u308c\u305f\u3053\u306e\u95a2\u6570\u306f\u3001\u8907\u7d20\u5909\u6570 s \u306b\u5bfe\u3057\u3066\u4ee5\u4e0b\u306e\u7d1a\u6570\u3067\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002 \u2003\u03b6(s) = \u2211_{n=1}^\u221e 1 \/ n^s<\/p>\n<p>\u5b9f\u969b\u306b\u306f\u5b9f\u90e8 Re(s) &gt; 1 \u306e\u9818\u57df\u3067\u3053\u306e\u7d1a\u6570\u304c\u53ce\u675f\u3057\u307e\u3059\u304c\u3001\u30ea\u30fc\u30de\u30f3\u306f\u89e3\u6790\u63a5\u7d9a\uff08analytic continuation\uff09\u306e\u624b\u6cd5\u3092\u7528\u3044\u3066 \u03b6(s) \u3092\u8907\u7d20\u5e73\u9762\u5168\u57df\u306b\u62e1\u5f35\u3057\u3001\u552f\u4e00\u306e\u6975\uff08simple pole\uff09\u3092 s = 1 \u306b\u6301\u3064\u8d85\u8d8a\u95a2\u6570\u3068\u3057\u3066\u5b9a\u7fa9\u3057\u307e\u3057\u305f\u3002\u3053\u306e\u89e3\u6790\u63a5\u7d9a\u3068\u3001\u4ee5\u4e0b\u306b\u793a\u3059\u6a5f\u80fd\u65b9\u7a0b\u5f0f\uff08functional equation\uff09\u304c\u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570\u306e\u6838\u5fc3\u3092\u6210\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n<p>\u2003\u2003\u039b(s) = \u03c0^{-s\/2} \u0393(s\/2) \u03b6(s) = \u039b(1 &#8211; s)<\/p>\n<p>\u3053\u3053\u3067 \u0393(s) \u306f\u30ac\u30f3\u30de\u95a2\u6570\u3067\u3059\u3002\u4e0a\u5f0f\u306f s \u3068 1\u2013s \u306e\u9593\u306b\u5bfe\u79f0\u6027\u3092\u4e0e\u3048\u3001\u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570\u306e\u89e3\u6790\u7684\u6027\u8cea\u3068\u6570\u8ad6\u7684\u5fdc\u7528\u3092\u5f37\u56fa\u306b\u7d50\u3073\u3064\u3051\u307e\u3059\u3002\u4ee5\u4e0b\u3001\u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570\u306e\u7279\u5fb4\u3068\u610f\u7fa9\u306b\u3064\u3044\u3066\u9806\u3092\u8ffd\u3063\u3066\u89e3\u8aac\u3057\u307e\u3059\u3002<\/p>\n<p>1. \u7d20\u6570\u3068\u306e\u6df1\u3044\u95a2\u4fc2\u6027 \u30ea\u30fc\u30de\u30f3\u306f\u30bc\u30fc\u30bf\u95a2\u6570\u306e\u7d1a\u6570\u8868\u793a\u3060\u3051\u3067\u306a\u304f\u3001\u30aa\u30a4\u30e9\u30fc\u7a4d\u8868\u793a\uff08Euler product\uff09\u3092\u3082\u793a\u3057\u307e\u3057\u305f\u3002 \u2003\u03b6(s) = \u220f_{p prime} (1 \u2013 p^{-s})^{-1} \u3053\u306e\u5f0f\u306f\u7d20\u6570 p \u3092\u6bcd\u4f53\u3068\u3057\u3066\u7121\u9650\u7a4d\u3092\u69cb\u6210\u3059\u308b\u3082\u306e\u3067\u3001\u30bc\u30fc\u30bf\u95a2\u6570\u3092\u901a\u3058\u3066\u7d20\u6570\u5206\u5e03\u3092\u89e3\u6790\u7684\u306b\u6271\u3046\u3053\u3068\u304c\u53ef\u80fd\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>2. \u30ea\u30fc\u30de\u30f3\u4e88\u60f3 \u30bc\u30fc\u30bf\u95a2\u6570\u306e\u300c\u975e\u81ea\u660e\u306a\u96f6\u70b9\uff08non-trivial zeros\uff09\u300d\u304c\u3059\u3079\u3066\u5b9f\u90e8 1\/2 \u4e0a\u306b\u5b58\u5728\u3059\u308b\u304b\u3069\u3046\u304b\u3001\u3068\u3044\u3046\u672a\u89e3\u6c7a\u554f\u984c\u304c\u30ea\u30fc\u30de\u30f3\u4e88\u60f3\u3067\u3059\u3002 \u2003Re(s) = 1\/2 \u3092\u6e80\u305f\u3059 s \u3067 \u03b6(s)=0 \u3053\u308c\u3092\u89e3\u6c7a\u3059\u308b\u3053\u3068\u306f\u7d20\u6570\u306e\u5206\u5e03\u306b\u95a2\u3059\u308b\u6700\u3082\u57fa\u672c\u7684\u306a\u554f\u3044\u3078\u306e\u300c\u9375\u300d\u3092\u4e0e\u3048\u308b\u3068\u8003\u3048\u3089\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n<p>3. \u89e3\u6790\u63a5\u7d9a\u3068\u6a5f\u80fd\u65b9\u7a0b\u5f0f \u03b6(s) \u306f\u3082\u3068\u3082\u3068 Re(s) &gt; 1 \u3067\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u304c\u3001\u89e3\u6790\u63a5\u7d9a\u306b\u3088\u3063\u3066\u8907\u7d20\u5168\u5e73\u9762\u3078\u62e1\u5f35\u3055\u308c\u307e\u3059\u3002\u5148\u8ff0\u306e\u6a5f\u80fd\u65b9\u7a0b\u5f0f\u306f\u305d\u306e\u62e1\u5f35\u5148\u3067\u6210\u308a\u7acb\u3061\u3001 s \u2194 1 \u2013 s \u306e\u5bfe\u79f0\u6027\u3092\u793a\u3057\u307e\u3059\u3002<\/p>\n<p>4. \u6570\u8ad6\u7684\u5fdc\u7528 \u30ea\u30fc\u30de\u30f3\u4e88\u60f3\u4ee5\u5916\u306b\u3082\u3001\u30bc\u30fc\u30bf\u95a2\u6570\u306f\u7d20\u6570\u5b9a\u7406\uff08Prime Number Theorem\uff09\u306e\u8a3c\u660e\u3084\u30c7\u30a3\u30ea\u30af\u30ec L \u95a2\u6570\u3001\u591a\u9805\u5f0f\u30bc\u30fc\u30bf\u95a2\u6570\u3001\u30bb\u30eb\u30d0\u30fc\u30b0\u306e\u30bc\u30fc\u30bf\u95a2\u6570\u306a\u3069\u6570\u591a\u304f\u306e\u5fdc\u7528\u5206\u91ce\u306b\u767a\u5c55\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n<p>5. \u7269\u7406\u5b66\u30fb\u7d71\u8a08\u529b\u5b66\u3078\u306e\u6ce2\u53ca \u91cf\u5b50\u7d71\u8a08\u529b\u5b66\u3084\u30e9\u30f3\u30c0\u30e0\u884c\u5217\u7406\u8ad6\u3068\u3082\u95a2\u9023\u304c\u6df1\u304f\u3001\u30bc\u30fc\u30bf\u95a2\u6570\u306e\u96f6\u70b9\u5206\u5e03\u306f\u91cf\u5b50\u529b\u5b66\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u6e96\u4f4d\u5206\u5e03\u3068\u6570\u5b66\u7684\u306b\u985e\u4f3c\u6027\u3092\u793a\u3059\u3053\u3068\u304c\u793a\u3055\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n<p>\u4ee5\u4e0a\u306e\u3088\u3046\u306b\u3001\u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570 \u03b6(s) \u306f\u89e3\u6790\u5b66\u30fb\u6570\u8ad6\u30fb\u7269\u7406\u5b66\u306e\u3055\u307e\u3056\u307e\u306a\u5206\u91ce\u3092\u3064\u306a\u3050\u300c\u30cf\u30d6\u300d\u3068\u3057\u3066\u6a5f\u80fd\u3057\u3001\u672a\u89e3\u6c7a\u306e\u30ea\u30fc\u30de\u30f3\u4e88\u60f3\u3092\u306f\u3058\u3081\u3068\u3059\u308b\u6570\u591a\u304f\u306e\u96e3\u554f\u3092\u62b1\u3048\u3066\u3044\u307e\u3059\u3002\u73fe\u5728\u3082\u4e16\u754c\u4e2d\u306e\u7814\u7a76\u8005\u304c\u305d\u306e\u6027\u8cea\u306e\u89e3\u660e\u306b\u6311\u307f\u3001\u8a3c\u660e\u3084\u8a08\u7b97\u6a5f\u5b9f\u9a13\u304c\u6d3b\u767a\u306b\u884c\u308f\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n<p>\u7279\u5fb4\u30ea\u30b9\u30c8 1. \u7d1a\u6570\u8868\u793a\uff1a\u03b6(s) = \u2211_{n=1}^\u221e 1\/n^s\uff08Re(s) &gt; 1\uff09 2. \u30aa\u30a4\u30e9\u30fc\u7a4d\u8868\u793a\uff1a\u03b6(s) = \u220f_{p prime} (1 \u2013 p^{-s})^{-1} 3. \u89e3\u6790\u63a5\u7d9a\uff1aRe(s) &gt; 1 \u306e\u9818\u57df\u304b\u3089\u8907\u7d20\u5e73\u9762\u5168\u57df\u3078 4. \u6a5f\u80fd\u65b9\u7a0b\u5f0f\uff1a\u039b(s) = \u03c0^{-s\/2} \u0393(s\/2) \u03b6(s) = \u039b(1 \u2013 s) 5. \u975e\u81ea\u660e\u96f6\u70b9\uff1aRe(s) = 1\/2 \u306b\u3042\u308b\u3068\u4e88\u60f3\uff08\u30ea\u30fc\u30de\u30f3\u4e88\u60f3\uff09 6. \u6570\u8ad6\u7684\u5fdc\u7528\uff1a\u7d20\u6570\u5b9a\u7406\u3084\u30c7\u30a3\u30ea\u30af\u30ec L \u95a2\u6570\u306a\u3069\u3078\u306e\u5c55\u958b 7. \u7269\u7406\u5b66\u3078\u306e\u5fdc\u7528\uff1a\u30e9\u30f3\u30c0\u30e0\u884c\u5217\u7406\u8ad6\u3084\u91cf\u5b50\u529b\u5b66\u3068\u306e\u95a2\u4fc2<\/p>\n<p>\u53c2\u8003\u6587\u732e\u30fb\u30a6\u30a7\u30d6\u30b5\u30a4\u30c8 1. \u5ca9\u6ce2\u8b1b\u5ea7\u300c\u73fe\u4ee3\u6570\u5b66\u306e\u5c55\u958b\u300d\u7b2c6\u5dfb \u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570\u8ad6\uff08\u5ca9\u6ce2\u66f8\u5e97\uff09 URL: <a href=\"https:\/\/www.iwanami.co.jp\/\">https:\/\/www.iwanami.co.jp\/<\/a> 2. \u6771\u4eac\u5927\u5b66\u6570\u7406\u79d1\u5b66\u7814\u7a76\u79d1\u300c\u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570\u3068\u305d\u306e\u5fdc\u7528\u300d\u8b1b\u7fa9\u8cc7\u6599 URL: <a href=\"http:\/\/www.ms.u-tokyo.ac.jp\/~lecture\/zeta\/\">http:\/\/www.ms.u-tokyo.ac.jp\/~lecture\/zeta\/<\/a> 3. Wolfram MathWorld \u201cRiemann Zeta Function\u201d URL: <a href=\"http:\/\/mathworld.wolfram.com\/RiemannZetaFunction.html\">http:\/\/mathworld.wolfram.com\/RiemannZetaFunction.html<\/a> \uff08\u82f1\u8a9e\u30fb\u65e5\u672c\u8a9e\u4e00\u90e8\u5bfe\u5fdc\uff09 4. \u4eac\u90fd\u5927\u5b66\u6570\u7406\u89e3\u6790\u7814\u7a76\u6240\u300c\u89e3\u6790\u63a5\u7d9a\u3068\u7279\u7570\u70b9\u300d URL: <a href=\"http:\/\/www.kurims.kyoto-u.ac.jp\/EMIS\/journals\/\">http:\/\/www.kurims.kyoto-u.ac.jp\/EMIS\/journals\/<\/a> 5. \u65e5\u672c\u6570\u5b66\u4f1a \u4f01\u753b\u9023\u8f09\u300c\u30ea\u30fc\u30de\u30f3\u4e88\u60f3\u5165\u9580\u300d URL: <a href=\"https:\/\/www.mathsoc.jp\/publication\/index.html\">https:\/\/www.mathsoc.jp\/publication\/index.html<\/a> 6. Wikipedia\u300c\u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570\u300d URL: <a href=\"https:\/\/ja.wikipedia.org\/wiki\/%E3%83%AA%E3%83%BC%E3%83%9E%E3%83%B3%E3%82%BC%E3%83%BC%E3%82%BF%E9%96%A2%E6%95%B0\">https:\/\/ja.wikipedia.org\/wiki\/\u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570<\/a> 7. Journal of Number Theory\uff08Elsevier\uff09\u201cRecent results on the Riemann Zeta Function\u201d URL: <a href=\"https:\/\/www.sciencedirect.com\/journal\/journal-of-number-theory\">https:\/\/www.sciencedirect.com\/journal\/journal-of-number-theory<\/a> \u4ee5\u4e0a\u306e\u8cc7\u6599\u3092\u53c2\u7167\u3059\u308b\u3068\u3001\u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570\u306e\u7406\u8ad6\u7684\u80cc\u666f\u304b\u3089\u6700\u65b0\u306e\u7814\u7a76\u52d5\u5411\u307e\u3067\u5e45\u5e83\u304f\u5b66\u3076\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p><a href=\"https:\/\/wordpress.realcollector.jp\/blog\/wp-content\/uploads\/2025\/12\/current_stablediffusion_api_response-281.png\" rel=\"attachment wp-att-65899\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"65899\" data-permalink=\"https:\/\/wordpress.realcollector.jp\/blog\/?attachment_id=65899\" 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data-large-file=\"https:\/\/wordpress.realcollector.jp\/blog\/wp-content\/uploads\/2025\/12\/current_stablediffusion_api_response-281.png\" src=\"https:\/\/wordpress.realcollector.jp\/blog\/wp-content\/uploads\/2025\/12\/current_stablediffusion_api_response-281.png\" alt=\"\" title=\"current_stablediffusion_api_response-281-png\" width=\"1024\" height=\"1024\" class=\"alignnone size-full wp-image-65899\" srcset=\"https:\/\/wordpress.realcollector.jp\/blog\/wp-content\/uploads\/2025\/12\/current_stablediffusion_api_response-281.png 512w, https:\/\/wordpress.realcollector.jp\/blog\/wp-content\/uploads\/2025\/12\/current_stablediffusion_api_response-281-300x300.png 300w, https:\/\/wordpress.realcollector.jp\/blog\/wp-content\/uploads\/2025\/12\/current_stablediffusion_api_response-281-150x150.png 150w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>zeta\uff08\u30bc\u30fc\u30bf\uff09\u3068\u306f\u3001\u4e3b\u306b\u6570\u5b66\u306e\u89e3\u6790\u7684\u624b\u6cd5\u3084\u6570\u8ad6\u306b\u304a\u3044\u3066\u767b\u5834\u3059\u308b\u91cd\u8981\u306a\u95a2\u6570\u3067\u3042\u308b\u300c\u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570\uff08Riemann Zeta Function\uff09\u300d\u3092\u6307\u3057\u307e\u3059\u3002\u30d9\u30eb\u30f3\u30cf\u30eb\u30c8\u30fb\u30ea\u30fc\u30de\u30f3\u306b\u3088\u3063\u30661859\u5e74\u306b\u5c0e\u5165\u3055\u308c\u305f\u3053\u306e\u95a2 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[10942],"tags":[14457,14324,14321,14323,14322],"class_list":["post-65898","post","type-post","status-publish","format-standard","hentry","category-o4-mini","tag-14457","tag-14324","tag-14321","tag-14323","tag-14322"],"jetpack_featured_media_url":"","jetpack-related-posts":[],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/wordpress.realcollector.jp\/blog\/index.php?rest_route=\/wp\/v2\/posts\/65898","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wordpress.realcollector.jp\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wordpress.realcollector.jp\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wordpress.realcollector.jp\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/wordpress.realcollector.jp\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=65898"}],"version-history":[{"count":1,"href":"https:\/\/wordpress.realcollector.jp\/blog\/index.php?rest_route=\/wp\/v2\/posts\/65898\/revisions"}],"predecessor-version":[{"id":65900,"href":"https:\/\/wordpress.realcollector.jp\/blog\/index.php?rest_route=\/wp\/v2\/posts\/65898\/revisions\/65900"}],"wp:attachment":[{"href":"https:\/\/wordpress.realcollector.jp\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=65898"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wordpress.realcollector.jp\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=65898"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wordpress.realcollector.jp\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=65898"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}