{"id":92599,"date":"2024-07-16T22:57:52","date_gmt":"2024-07-16T22:57:52","guid":{"rendered":"https:\/\/science-hub.click\/%E3%82%AA%E3%82%A4%E3%83%A9%E3%83%BC%E7%A9%8D%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E8%A9%B3%E3%81%97%E3%81%8F%E8%A7%A3%E8%AA%AC\/"},"modified":"2024-07-16T22:57:52","modified_gmt":"2024-07-16T22:57:52","slug":"%E3%82%AA%E3%82%A4%E3%83%A9%E3%83%BC%E7%A9%8D%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E8%A9%B3%E3%81%97%E3%81%8F%E8%A7%A3%E8%AA%AC","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=92599","title":{"rendered":"\u30aa\u30a4\u30e9\u30fc\u7a4d\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac"},"content":{"rendered":"<div><div><h2>\u5c0e\u5165<\/h2><div><div><figure class=\"wp-block-image size-large is-style-default\">\n<img decoding=\"async\" alt=\"\" class=\"aligncenter\" onerror=\"this.style.display=none;\" src=\"https:\/\/img.youtube.com\/vi\/rYf8I-D-oo0\/0.jpg\" style=\"width:100%;\"\/><\/figure><div>\u30ec\u30aa\u30f3\u30cf\u30eb\u30c8\u30fb\u30aa\u30a4\u30e9\u30fc<\/div><\/div><\/div><p>\u6570\u5b66\u3001\u3088\u308a\u6b63\u78ba\u306b\u306f\u89e3\u6790\u7684\u6574\u6570\u8ad6\u3067\u306f\u3001<b>\u30aa\u30a4\u30e9\u30fc\u7a4d\u306f<\/b>\u3001\u7d20\u6570\u306b\u3088\u3063\u3066\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u4ed8\u3051\u3055\u308c\u305f\u7121\u9650\u7a4d\u3078\u306e\u62e1\u5f35\u3067\u3059\u3002<\/p><p>\u3053\u308c\u306b\u3088\u308a\u7d20\u6570\u306e\u5206\u5e03\u306e<i>\u6e2c\u5b9a<\/i>\u304c\u53ef\u80fd\u306b\u306a\u308a\u3001 <span><a href=\"https:\/\/science-hub.click\/?p=15788\">\u30ea\u30fc\u30de\u30f3\u306e\u30bc\u30fc\u30bf\u95a2\u6570<\/a><\/span>\u3068\u5bc6\u63a5\u306b\u95a2\u9023\u3057\u3066\u3044\u307e\u3059\u3002<\/p><p>\u30b9\u30a4\u30b9\u306e<span><a href=\"https:\/\/science-hub.click\/?p=17102\">\u6570\u5b66\u8005<\/a><\/span><span><a href=\"https:\/\/science-hub.click\/?p=48545\">\u30ec\u30aa\u30f3\u30cf\u30eb\u30c8\u30fb\u30aa\u30a4\u30e9\u30fc<\/a><\/span>\u306b\u3061\u306a\u3093\u3067\u540d\u4ed8\u3051\u3089\u308c\u307e\u3057\u305f\u3002<\/p><figure class=\"wp-block-image size-large is-style-default\">\n<img decoding=\"async\" alt=\"\u30aa\u30a4\u30e9\u30fc\u7a4d\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\" class=\"aligncenter\" onerror=\"this.style.display=none;\" src=\"https:\/\/img.youtube.com\/vi\/MxhpG5yAv2A\/0.jpg\" style=\"width:100%;\"\/><\/figure><h2>\u305d\u306e\u4ed6\u306e\u30aa\u30a4\u30e9\u30fc\u88fd\u54c1<\/h2><h3><span><span><a href=\"https:\/\/science-hub.click\/?p=41860\">\u30c7\u30a3\u30ea\u30af\u30ec\u306e\u30ad\u30e3\u30e9\u30af\u30bf\u30fc<\/a><\/span><\/span><\/h3><p>\u30c7\u30a3\u30ea\u30af\u30ec\u306f\u3001 <i>m<\/i>\u3068<i>n<\/i>\u304c\u4e92\u3044\u306b\u7d20\u3067\u3042\u308b\u5834\u5408\u3001 <i>Z<\/i> \/ <i>nZ<\/i>\u306e\u30af\u30e9\u30b9<i>m<\/i>\u306e\u7d20\u6570\u306e\u6570\u306f<span><a href=\"https:\/\/science-hub.click\/?p=96157\">\u7121\u9650<\/a><\/span>\u3067<span><a href=\"https:\/\/science-hub.click\/?p=71097\">\u3042\u308b<\/a><\/span>\u3053\u3068\u3092\u8a3c\u660e\u3057\u305f\u3044\u3068\u8003\u3048\u3066\u3044\u307e\u3059\u3002\u5f7c\u306f\u73fe\u5728\u81ea\u5206\u306e\u540d\u524d\u3092\u51a0\u3057\u3066\u3044\u308b\u6587\u5b57\u3092\u4f7f\u7528\u3057\u3066\u304a\u308a\u3001\u8a73\u7d30\u8a18\u4e8b\u306e<span><a href=\"https:\/\/science-hub.click\/?p=92599\">\u30aa\u30a4\u30e9\u30fc\u7a4d\u306e<\/a><\/span>\u6bb5\u843d\u3067\u8aac\u660e\u3055\u308c\u3066\u3044\u308b\u8a08\u7b97\u4e2d\u306b\u3001\u6b21\u306e\u7a4d\u304c\u5f97\u3089\u308c\u307e\u3059\u3002 <\/p><center><div class=\"math-formual notranslate\">$$ {\\prod_{p \\in \\mathcal P} \\Big(1 -\\frac {\\chi(p)}{p^s}\\Big)^{-1}} $$<\/div><\/center><p>\u3053\u3053\u3067\u3001\u03c7 \u306f\u30c7\u30a3\u30ea\u30af\u30ec\u6587\u5b57\u3092\u793a\u3057\u3001\u6587\u5b57\u306e<span><a href=\"https:\/\/science-hub.click\/?p=57227\">\u96c6\u5408\u306f<\/a><\/span>\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059\u3002 <div class=\"math-formual notranslate\">$$ {\\scriptstyle \\widehat U} $$<\/div> <i>s \u306f<\/i>\u53b3\u5bc6\u306b 1 \u3088\u308a\u5927\u304d\u3044<span><a href=\"https:\/\/science-hub.click\/?p=93137\">\u5b9f\u6570<\/a><\/span>\u3092\u8868\u3057\u307e\u3059\u3002\u6b21\u306b\u3001\u30c7\u30a3\u30ea\u30af\u30ec\u306f\u30aa\u30a4\u30e9\u30fc\u7a4d\u306e\u30d5\u30a1\u30df\u30ea\u30fc\u3092\u78ba\u7acb\u3057\u307e\u3059\u3002 <\/p><center><div class=\"math-formual notranslate\">$$ {\\forall s \\in ]1, +\\infty[ \\quad \\forall \\chi \\in \\widehat U \\quad L(s, \\chi) = \\sum_{k=1}^{\\infty} \\frac {\\chi(k)}{k^s} \\ = \\ \\prod_{p \\in \\mathcal P} \\Big(1 -\\frac {\\chi(p)}{p^s}\\Big)^{-1} } $$<\/div><\/center><p>\u5b9f\u969b\u3001\u95a2\u6570 \u03c7 \u306f<span>\u5b8c\u5168\u306b<\/span>\u4e57\u7b97\u7684\u3067\u3042\u308a\u3001\u30aa\u30a4\u30e9\u30fc\u8a08\u7b97\u3082\u540c\u69d8\u306b\u9069\u7528\u3055\u308c\u307e\u3059\u3002<\/p><dl><dd><ul><li>\u95a2\u6570<i>L<\/i> ( <i>s<\/i> , \u03c7) \u306f\u3001\u6587\u5b57 \u03c7 \u306e<b>\u30c7\u30a3\u30ea\u30af\u30ec L \u7d1a\u6570<\/b>\u3068\u547c\u3070\u308c\u307e\u3059\u3002<\/li><\/ul><\/dd><\/dl><p> <i>s<\/i>\u304c\u5b9f\u6570\u90e8 &gt; 1 \u306e<span><a href=\"https:\/\/science-hub.click\/?p=94201\">\u8907\u7d20\u6570<\/a><\/span>\u306e\u5834\u5408\u3001<span>\u53ce\u675f\u306f<\/span><span><a href=\"https:\/\/science-hub.click\/?p=108705\">\u7d76\u5bfe\u7684<\/a><\/span>\u3067\u3059\u3002\u89e3\u6790\u7684\u62e1\u5f35\u306b\u3088\u308a\u3001\u3053\u306e\u95a2\u6570\u306f<span><a href=\"https:\/\/science-hub.click\/?p=46336\">\u8907\u7d20\u5e73\u9762<\/a><\/span>\u5168\u4f53\u4e0a\u306e\u6709\u7406\u578b\u95a2\u6570\u306b\u62e1\u5f35\u3067\u304d\u307e\u3059\u3002<\/p><p>\u30c7\u30a3\u30ea\u30af\u30ec L \u7d1a\u6570\u306f\u30ea\u30fc\u30de\u30f3 \u30bc\u30fc\u30bf\u95a2\u6570\u306e\u76f4\u63a5\u4e00\u822c\u5316\u3067\u3042\u308a\u3001<span>\u4e00\u822c\u5316\u3055\u308c\u305f\u30ea\u30fc\u30de\u30f3\u4e88\u60f3<\/span>\u306e\u4e2d\u3067\u9855\u8457\u3067\u3042\u308b\u3068\u601d\u308f\u308c\u307e\u3059\u3002<\/p><h3><span><span><a href=\"https:\/\/science-hub.click\/?p=7924\">\u4e00\u822c\u5316<\/a><\/span><\/span><\/h3><p>\u4e00\u822c\u306b\u3001\u6b21\u306e\u5f62\u5f0f\u306e\u30c7\u30a3\u30ea\u30af\u30ec\u7d1a\u6570<\/p><dl><dd><div class=\"math-formual notranslate\">$$ {\\sum_{n} a(n)n^{-s}\\,} $$<\/div><\/dd><\/dl><p>\u307e\u305f\u306f<div class=\"math-formual notranslate\">$$ {a(n)\\,} $$<\/div>\u306f<i>n<\/i>\u306e\u4e57\u7b97\u95a2\u6570\u3067\u3042\u308a\u3001\u6b21\u306e\u5f62\u5f0f\u3067\u8a18\u8ff0\u3067\u304d\u307e\u3059\u3002 <\/p><dl><dd><div class=\"math-formual notranslate\">$$ {\\prod_{p} P(p,s)\\,} $$<\/div><\/dd><\/dl><p>\u307e\u305f\u306f<div class=\"math-formual notranslate\">$$ {P(p,s)\\,} $$<\/div>\u5408\u8a08\u3067\u3059<\/p><dl><dd><div class=\"math-formual notranslate\">$$ {1 + a(p)p^{-s} + a(p^2)p^{-2s} + \\ldots\\,} $$<\/div> \u3002<\/dd><\/dl><p>\u5b9f\u969b\u3001\u3053\u308c\u3089\u3092\u5f62\u5f0f\u7684\u306a\u751f\u6210\u95a2\u6570\u3068\u8003\u3048\u308b\u3068\u3001\u305d\u306e\u3088\u3046\u306a<i>\u5f62\u5f0f\u7684\u306a<\/i>\u30aa\u30a4\u30e9\u30fc\u7a4d\u5c55\u958b\u306e\u5b58\u5728\u306f\u3001 <div class=\"math-formual notranslate\">$$ {a(n)\\,} $$<\/div>\u306f\u4e57\u6cd5\u7684\u3067\u3059: \u3053\u308c\u306f\u307e\u3055\u306b\u305d\u308c\u3092\u793a\u3057\u3066\u3044\u307e\u3059<div class=\"math-formual notranslate\">$$ {a(n)\\,} $$<\/div>\u306e\u88fd\u54c1\u3067\u3059<div class=\"math-formual notranslate\">$$ {a(p^k)\\,} $$<\/div> <i>n \u304c<\/i>\u3079\u304d\u4e57\u306e\u7a4d\u3092\u56e0\u6570\u5206\u89e3\u3059\u308b\u3068\u304d<div class=\"math-formual notranslate\">$$ {p^k\\,} $$<\/div>\u7570\u306a\u308b\u7d20\u6570 p.<\/p><p>\u5b9f\u969b\u306b\u306f\u3001\u3059\u3079\u3066\u306e\u91cd\u8981\u306a\u30b1\u30fc\u30b9\u306f\u3001\u7121\u9650\u7d1a\u6570\u3068\u7121\u9650\u7a4d\u5c55\u958b\u304c\u7279\u5b9a\u306e\u9818\u57df\u3067\u7d76\u5bfe\u306b\u53ce\u675f\u3059\u308b\u3088\u3046\u306a\u3082\u306e\u3067\u3059\u3002<\/p><dl><dd> <span><i>R<\/i> <i>e<\/i> ( <i>s<\/i> ) &gt; <i>C<\/i><\/span> :<\/dd><\/dl><p>\u3064\u307e\u308a\u3001\u7279\u5b9a\u306e\u53f3\u534a\u5e73\u9762\u306e\u8907\u7d20\u6570\u306b\u304a\u3044\u3066\u3067\u3059\u3002\u7121\u9650\u7a4d\u304c\u53ce\u675f\u3059\u308b\u306b\u306f\u30bc\u30ed<span><a href=\"https:\/\/science-hub.click\/?p=28052\">\u4ee5\u5916<\/a><\/span>\u306e\u5024\u3092\u4e0e\u3048\u308b\u5fc5\u8981\u304c\u3042\u308b\u305f\u3081\u3001\u3053\u308c\u3067\u3059\u3067\u306b\u3042\u308b\u7a0b\u5ea6\u306e\u60c5\u5831\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u7121\u9650\u7d1a\u6570\u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u308b\u95a2\u6570\u306f\u3001\u305d\u306e\u3088\u3046\u306a\u534a\u5e73\u9762\u3067\u306f<span><a href=\"https:\/\/science-hub.click\/?p=5522\">\u30bc\u30ed<\/a><\/span>\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/p><p>\u91cd\u8981\u306a\u7279\u6b8a\u306a\u30b1\u30fc\u30b9\u3068\u3057\u3066\u306f\u3001 <div class=\"math-formual notranslate\">$$ {P(p,s)\\,} $$<\/div>\u306f\u5e7e\u4f55\u7d1a\u6570\u3067\u3059\u3002 <div class=\"math-formual notranslate\">$$ {a(n)\\,} $$<\/div>\u306f\u5b8c\u5168\u306b\u4e57\u7b97\u7684\u3067\u3059\u3002\u305d\u308c\u3067\u306f\u3001 <\/p><dl><dd><div class=\"math-formual notranslate\">$$ {P(p,s) = \\frac{1}{1 &#8211; a(p)p^{-s}}\\,} $$<\/div><\/dd><\/dl><p>\u30ea\u30fc\u30de\u30f3\u30bc\u30fc\u30bf\u95a2\u6570\u306e\u5834\u5408\u3068\u540c\u69d8\u3067\u3059\uff08 <div class=\"math-formual notranslate\">$$ {a(n) = 1\\,} $$<\/div> \uff09\u3001\u3088\u308a\u4e00\u822c\u7684\u306b\u306f\u30c7\u30a3\u30ea\u30af\u30ec\u306e\u30ad\u30e3\u30e9\u30af\u30bf\u30fc\u306b\u5f53\u3066\u306f\u307e\u308a\u307e\u3059\u3002\u30e2\u30b8\u30e5\u30e9\u30fc\u5f62\u5f0f\u306e\u7406\u8ad6\u3067\u306f\u3001\u5206\u6bcd\u3068\u3057\u3066 2 \u6b21\u591a\u9805\u5f0f\u3092\u4f7f\u7528\u3057\u305f\u30aa\u30a4\u30e9\u30fc\u7a4d\u3092\u4f7f\u7528\u3059\u308b\u306e\u304c\u4e00\u822c\u7684\u3067\u3059\u3002\u4e00\u822c\u7684\u306a\u30e9\u30f3\u30b0\u30e9\u30f3\u30ba \u30d7\u30ed\u30b0\u30e9\u30e0\u306b\u306f\u3001 <i>m<\/i><span>\u6b21<\/span>\u306e\u591a\u9805\u5f0f\u306e\u63a5\u7d9a\u3068\u6b21\u6570\u306e\u8868\u73fe\u7406\u8ad6\u306e\u6bd4\u8f03\u8aac\u660e\u304c\u542b\u307e\u308c\u3066\u3044\u307e\u3059\u3002 <div class=\"math-formual notranslate\">$$ {GL_m\\,} $$<\/div> \u3002<\/p><h2>\u30aa\u30a4\u30e9\u30fc\u306e\u4f5c\u54c1<\/h2><figure class=\"wp-block-image size-large is-style-default\">\n<img decoding=\"async\" alt=\"\u30aa\u30a4\u30e9\u30fc\u7a4d\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\" class=\"aligncenter\" onerror=\"this.style.display=none;\" src=\"https:\/\/img.youtube.com\/vi\/6b7Lr3lVCII\/0.jpg\" style=\"width:100%;\"\/><\/figure><h3><span>\u30aa\u30a4\u30e9\u30fc\u5fae\u7a4d\u5206<\/span><\/h3><p>\u30aa\u30a4\u30e9\u30fc\u306f\u7d20\u6570\u306e\u5206\u5e03\u3092\u8a55\u4fa1\u3057\u3088\u3046\u3068\u3057\u307e\u3059\u3002\u7d20\u6570\u306e\u96c6\u5408\u306f\u3053\u3053\u3067\u306f<i>P<\/i>\u3067\u8868\u3055\u308c\u307e\u3059\u3002\u3053\u306e\u305f\u3081\u306b\u3001\u5f7c\u306f\u6b21\u306e\u5f0f\u3092\u78ba\u7acb\u3057\u307e\u3057\u305f\u3002 <\/p><div class=\"math-formual notranslate\">$$ { \\forall s \\in \\mathbb C \\quad \\mathfrak {Re} (s) &gt; 1 \\Rightarrow \\sum_{n=1}^\\infin \\ \\frac{1}{n^s} \\ = \\ \\prod_{p\\in\\mathcal{P}} \\ \\frac{1}{1-p^{-s}}} $$<\/div><p>\u3053\u3053\u3067\u3001 <i>Re<\/i> ( <i>s<\/i> ) \u306f<i>s<\/i>\u306e\u5b9f\u90e8\u3092\u8868\u3057\u307e\u3059\u3002<\/p><p>\u30aa\u30a4\u30e9\u30fc\u306f\u5de6\u5074\u306e\u9805\u306b<span><a href=\"https:\/\/science-hub.click\/?p=43514\">\u30bc\u30fc\u30bf\u95a2\u6570\u3068\u3044\u3046<\/a><\/span>\u540d\u524d\u3092\u4e0e\u3048\u3001\u8907\u7d20\u534a\u5e73\u9762\u4e0a\u3067\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002 <\/p><div class=\"math-formual notranslate\">$$ {\\forall s \\in \\mathbb C \\quad \\mathfrak {Re} (s) &gt; 1 \\quad \\zeta(s) \\ = \\ \\sum_{n=1}^\\infin \\ \\frac{1}{n^s}} $$<\/div><p>\u3053\u306e\u95a2\u6570\u306f\u3001\u8907\u7d20\u5e73\u9762\u5168\u4f53\u306b\u308f\u305f\u3063\u3066\u89e3\u6790\u7684\u306b\u62e1\u5f35\u3055\u308c\u3001\u6709\u7406\u578b\u95a2\u6570\u306b\u306a\u308a\u307e\u3059\u3002<\/p><div align=\"left\"><div title=\"[\u62e1\u5927\u3059\u308b]\"><div align=\"left\"><p> <i>k \u3092<\/i>\u53b3\u5bc6\u306b\u6b63\u306e\u6574\u6570\u3001 <i>P<\/i> <sub>k \u3092<\/sub>\u6700\u521d\u306e<i>k<\/i>\u500b\u306e\u7d20\u6570\u306e\u30bb\u30c3\u30c8\u3001 <i>N<\/i> <sub>k \u3092<\/sub>\u53b3\u5bc6\u306b\u6b63\u306e\u6574\u6570\u306e\u30bb\u30c3\u30c8\u3068\u3057\u3001\u305d\u306e\u7d20\u56e0\u6570\u3078\u306e<span><a href=\"https:\/\/science-hub.click\/?p=4434\">\u5206\u89e3<\/a><\/span>\u306b\u306f\u96c6\u5408<i>P<\/i> <sub>k<\/sub>\u304b\u3089\u306e\u7d20\u6570\u306e\u307f\u304c\u542b\u307e\u308c\u307e\u3059\u3002 <i>P<\/i> <sub>k<\/sub>\u306e\u8981\u7d20\u306f\u3001 <i>p<\/i> <sub>1<\/sub> \u3001&#8230;\u3001 <i>p<\/i> <sub>k<\/sub>\u3067\u8868\u3055\u308c\u307e\u3059\u3002\u6574\u6570<i>n<\/i>\u306e\u7d20\u56e0\u6570\u5206\u89e3\u306e\u6700\u5927<span><a href=\"https:\/\/science-hub.click\/?p=42582\">\u6307\u6570<\/a><\/span><i>\u306f E<\/i> ( <i>n<\/i> ) \u3067\u8868\u3055\u308c\u307e\u3059\u3002<\/p><p>\u3053\u3053\u3067\u306e\u8868\u8a18\u03b1\u306f\u3001\u6b63\u306e\u6574\u6570\u306e<i>\uff4b<\/i>\u30bf\u30d7\u30eb\uff08\u03b1 <sub>\uff11<\/sub> \u3001\u03b1 <sub>\uff12<\/sub> \u3001\u2026\u3001\u03b1 <sub>\uff4b<\/sub> \uff09\u3092\u793a\u3057\u3001\uff2e\uff08\u03b1\uff09\u306f\u3001 <i>\uff4b<\/i>\u30bf\u30d7\u30eb\u304c\u5230\u9054\u3059\u308b\u6700\u5927\u5024\u3092\u793a\u3059\u3002<\/p><p>\u6700\u5f8c\u306b\u3001 <i>s \u306f<\/i>\u5b9f\u90e8\u304c\u53b3\u5bc6\u306b 1 \u3088\u308a\u5927\u304d\u3044\u8907\u7d20\u6570\u3092\u793a\u3057\u3001 <i>l \u306f<\/i>\u53b3\u5bc6\u306b\u6b63\u306e\u6574\u6570\u3092\u793a\u3057\u307e\u3059\u3002\u76ee\u7684\u306f\u3001\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u5408\u8a08<i>S<\/i> <sub>kl<\/sub> (s) \u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u3067\u3059\u3002 <\/p><center><div class=\"math-formual notranslate\">$$ { S_{kl}(s) \\ = \\ \\sum_{n \\in N_k \\ E(n) \\le l} \\frac 1{n^s} } $$<\/div><\/center><p>\u6700\u521d\u306b<i>l<\/i>\u3067\u3001\u6b21\u306b<i>k<\/i>\u3067\u9650\u754c\u307e\u3067 2 \u56de\u901a\u904e\u3059\u308b\u3068\u3001\u7d50\u8ad6\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u5b9f\u969b\u3001\u5408\u8a08\u3082\u6b21\u306e\u3088\u3046\u306b\u66f8\u304b\u308c\u3066\u3044\u308b\u3053\u3068\u306b\u6c17\u4ed8\u304d\u307e\u3059\u3002 <\/p><center><div class=\"math-formual notranslate\">$$ { (1) \\quad S_{kl}(s) \\ = \\ \\sum_{N(\\alpha) \\le l} \\frac 1{(p_1^s)^{\\alpha_1}.(p_2^s)^{\\alpha_2}. \\cdots .(p_k^s)^{\\alpha_k}} \\ = \\ \\sum_{N(\\alpha) \\le l} \\ \\prod_{i = 1}^k \\frac 1{(p_i^s)^{\\alpha_i}} = \\prod_{i = 1}^k  \\ \\sum_{j = 0}^l \\frac 1{(p_i^s)^j}} $$<\/div><\/center><p>\u5f97\u3089\u308c\u305f\u7a4d\u306e<i>k<\/i>\u500b\u306e\u5408\u8a08\u306f\u305d\u308c\u305e\u308c\u7d76\u5bfe\u306b\u53ce\u675f\u3057\u307e\u3059\u3002\u79c1\u305f\u3061\u306f\u6b21\u306e\u3088\u3046\u306b\u63a8\u6e2c\u3057\u307e\u3059\u3002 <\/p><center><div class=\"math-formual notranslate\">$$ {(2) \\quad \\sum_{n \\in N_k \\ E(k) \\le l} \\frac 1{|n^s|}  \\le \\prod_{i = 1}^k  \\ \\sum_{j = 0}^{\\infty} \\frac 1{|p_i^s|^j} \\ = \\ \\prod_{i = 1}^k \\frac 1{1-|p_i^{-s}|}} $$<\/div><\/center><p>\u3057\u305f\u304c\u3063\u3066\u3001\u5897\u52a0\u306e<i>l<\/i>\u306e\u7d1a\u6570<i>(2)<\/i>\u306f\u7d76\u5bfe\u306b\u53ce\u675f\u3057\u3001\u6b21\u306e\u7b49\u5f0f\u304c\u63a8\u5b9a\u3055\u308c\u307e\u3059\u3002 <\/p><center><div class=\"math-formual notranslate\">$$ {(3) \\quad \\sum_{n \\in N_k} \\frac 1{n^s} \\ = \\ \\prod_{i = 1}^k \\frac 1{1-p_i^{-s}}} $$<\/div><\/center><p>\u7b49\u5f0f<i>(3)<\/i>\u306e<i>k<\/i>\u306e\u7d1a\u6570\u3082\u7d76\u5bfe\u53ce\u675f\u3057\u3066\u304a\u308a\u3001\u6b21\u306e\u3088\u3046\u306b\u63a8\u6e2c\u3055\u308c\u307e\u3059\u3002 <\/p><div class=\"math-formual notranslate\">$$ {\\forall s \\in \\mathbb C \\quad \\mathfrak {Re} (s) &gt; 1 \\Rightarrow \\zeta(s) \\ = \\ \\sum_{n=1}^\\infin \\ \\frac{1}{n^s} \\ = \\ \\prod_{i = 1}^{\\infty} \\frac 1{1-p_i^{-s}}} $$<\/div><\/div><\/div><\/div><h3><span>\u7d20\u6570\u306e\u6700\u521d\u306e\u5206\u5e03<\/span><\/h3><p>\u76ee\u7684\u306f\u3001\u7d20\u6570\u306e<span><a href=\"https:\/\/science-hub.click\/?p=21808\">\u983b\u5ea6<\/a><\/span>\u306b\u95a2\u3059\u308b\u7b2c\u4e00\u6cd5\u5247\u3092\u6c7a\u5b9a\u3059\u308b\u3053\u3068\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u305f\u3068\u3048\u3070\u3001\u5b8c\u5168\u306a\u6b63\u65b9\u5f62\u3088\u308a\u591a\u3044\u304b\u5c11\u306a\u3044\u304b\u3068\u3044\u3046\u8cea\u554f\u306b\u7b54\u3048\u308b\u3053\u3068\u304c\u53ef\u80fd\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u547d\u984c\u306f\u3001 <i>N<\/i>\u304c\u5341\u5206\u306b\u5927\u304d\u3044\u6574\u6570\u3067\u3042\u308b\u5834\u5408\u3001 <i>N<\/i>\u3088\u308a\u5c0f\u3055\u3044\u5b8c\u5168\u5e73\u65b9\u306f\u5b58\u5728\u3059\u308b\u304b\u3001\u307e\u305f\u306f\u305d\u308c\u3088\u308a\u5c0f\u3055\u3044\u304b\u3068\u3044\u3046<span><a href=\"https:\/\/science-hub.click\/?p=81037\">\u610f\u5473<\/a><\/span>\u3067\u8aad\u307e\u308c\u306a\u3051\u308c\u3070\u306a\u308a\u307e\u305b\u3093\u3002\u30aa\u30a4\u30e9\u30fc\u306f\u3001\u6b21\u306e\u6570\u5217\u306e\u767a\u6563\u3092\u8a3c\u660e\u3059\u308b\u3053\u3068\u3067\u3053\u306e\u8cea\u554f\u306b\u7b54\u3048\u307e\u3059\u3002 <\/p><center><div class=\"math-formual notranslate\">$$ {\\sum_{p \\in \\mathcal P} \\frac 1p\\ = \\ + \\infty } $$<\/div><\/center><p>\u3057\u305f\u304c\u3063\u3066\u3001<span><a href=\"https:\/\/science-hub.click\/?p=95765\">\u3059\u3079\u3066\u306e<\/a><\/span><i>n<\/i>\u306b\u3064\u3044\u3066\u3001\u7d20\u6570\u306e\u6570\u304c\u5b8c\u5168\u5e73\u65b9\u306e\u6570\u3088\u308a\u3082\u591a\u304f\u306a\u308b\u3088\u3046\u306b<i>n<\/i>\u3088\u308a\u5927\u304d\u3044\u6570<i>N<\/i>\u304c\u5b58\u5728\u3059\u308b\u5834\u5408\u3001\u4e00\u822c\u9805 1\/ <i>n<\/i> <sup>2<\/sup>\u306e\u7cfb\u5217\u306f\u200b\u200b\u767a\u6563\u3057\u307e\u3059\u304c\u3001\u305d\u3046\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/p><p>\u76ee\u7684\u306f\u3001\u4e00\u9023\u306e\u7d20\u6570\u306b\u76f8\u5f53\u3059\u308b\u3082\u306e\u3092\u898b\u3064\u3051\u308b\u3053\u3068\u3067\u3059\u3002\u305d\u308c\u306f<span><a href=\"https:\/\/science-hub.click\/?p=11116\">\u7d20\u6570\u5b9a\u7406<\/a><\/span>\u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002<\/p><div align=\"left\"><div title=\"[\u62e1\u5927\u3059\u308b]\"><div align=\"left\"><p>\u6b21\u306e\u7b49\u5f0f\u306f\u3001 <i>s \u304c<\/i>1 \u306b\u306a\u308b\u50be\u5411\u304c\u3042\u308b\u5834\u5408\u3001\u95a2\u6570 \u03b6 \u304c\u767a\u6563\u3059\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u307e\u3059\u3002 <\/p><div class=\"math-formual notranslate\">$$ {\\forall s &gt; 1 \\quad \\zeta(s) = \\sum_{n=1}^{\\infty}\\frac 1{n^s}} $$<\/div><p>\u3053\u3053\u3067<span>ln \u306f<\/span><span>\u81ea\u7136\u5bfe\u6570<\/span>\u3092\u8868\u3057\u307e\u3059\u3002<span>\u5bfe\u6570<\/span>\u95a2\u6570\u306e\u51f9\u9762\u306f\u6b21\u306e\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u307e\u3059\u3002 <\/p><div class=\"math-formual notranslate\">$$ {\\forall x \\in ]0,1-e^{-1}[ \\quad x &gt; -(1-e^{-1})\\ln (1-x) } $$<\/div><p> <i>p<\/i>\u304c<span><a href=\"https:\/\/science-hub.click\/?p=16478\">\u7d20\u6570<\/a><\/span>\u306e\u5834\u5408\u3001\u305d\u308c\u306f (1 &#8211; e <sup>-1<\/sup> ) <sup>-1<\/sup>\u3088\u308a\u5927\u304d\u304f\u3001\u304b\u3064\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002 <\/p><div class=\"math-formual notranslate\">$$ {\\forall s &gt; 1,\\; \\forall p \\in \\mathcal P \\quad p^{-s} &gt;  -(1-e^{-1})\\ln (1-p^{-s})} $$<\/div><p>\u30aa\u30a4\u30e9\u30fc\u7a4d\u306b\u3088\u308a\u3001\u6b21\u306e\u5897\u52a0\u3092\u63a8\u5b9a\u3067\u304d\u307e\u3059\u3002 <\/p><div class=\"math-formual notranslate\">$$ {(1)\\quad \\sum_{p \\in \\mathcal P} p^{-s} &gt; (1-e^{-1})^{-1} \\ln \\left(\\prod_{p \\in \\mathcal P} \\frac 1{1 &#8211; p^{-s}}\\right) = (1-e^{-1})^{-1} \\ln(\\zeta (s))} $$<\/div><p>\u5897\u52a0<i>(1)<\/i>\u306f\u3001 <i>s \u304c<\/i>1 \u306b\u5411\u304b\u3046\u50be\u5411\u304c\u3042\u308b\u5834\u5408\u3001\u53f3\u5074\u306e\u9805\u306f\u7121\u9650\u5927\u306b\u5411\u304b\u3046\u50be\u5411\u304c\u3042\u308a\u3001\u3057\u305f\u304c\u3063\u3066\u5de6\u5074\u306e\u9805\u3082\u7121\u9650\u5927\u306b\u5411\u304b\u3046\u50be\u5411\u304c\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u304a\u308a\u3001\u547d\u984c\u3092\u793a\u3057\u3066\u3044\u307e\u3059\u3002<\/p><\/div><\/div><\/div><h3> <span><i>s \u304c<\/i>2 \u306b\u7b49\u3057\u3044\u5834\u5408\u306e\u8a08\u7b97<\/span><\/h3><p>\u30aa\u30a4\u30e9\u30fc\u306f\u3001 <i>s \u304c<\/i>2 \u306b\u7b49\u3057\u3044\u5834\u5408\u306e\u95a2\u6570 \u03b6 \u306e\u5024\u3092\u6c7a\u5b9a\u3059\u308b\u3053\u3068\u306b\u6210\u529f\u3057\u307e\u3057\u305f\u3002\u8a08\u7b97\u306f<span><a href=\"https:\/\/science-hub.click\/?p=39212\">\u9ad8\u8abf\u6ce2<\/a><\/span>\u89e3\u6790\u30c4\u30fc\u30eb\u3092\u4f7f\u7528\u3059\u308b\u3053\u3068\u3067\u975e\u5e38\u306b\u7c21\u5358\u306b\u5f97\u3089\u308c\u307e\u3059\u3002\u3053\u308c\u3092\u884c\u3046\u306b\u306f\u3001<span><a href=\"https:\/\/science-hub.click\/?p=92849\">\u30d1\u30fc\u30bb\u30f4\u30a1\u30eb\u306e\u7b49\u5f0f\u3092<\/a><\/span>\u3001\u5468\u671f 2\u03c0 \u306e<i>f<\/i>\u3067\u793a\u3055\u308c\u308b\u5468\u671f\u95a2\u6570\u306e<span>\u30d5\u30fc\u30ea\u30a8\u5909\u63db<\/span>\u306b\u9069\u7528\u3057\u3001[-\u03c0, \u03c0[ \u306e\u6052\u7b49\u5f0f\u306b\u7b49\u3057\u3044] \u3092\u9069\u7528\u3059\u308b\u3060\u3051\u3067\u5341\u5206\u3067\u3059\u3002\u4ee5\u4e0b\u3092\u53d6\u5f97\u3057\u307e\u3059\u3002 <\/p><center><div class=\"math-formual notranslate\">$$ {\\zeta (2) \\ = \\ \\prod_{p \\ \\in P} \\frac 1{1- p^{-2}} \\ = \\ \\sum_{n=1}^{+\\infty} \\frac1{n^2} = \\frac {\\pi^2}6 } $$<\/div><\/center><p>\u3053\u306e\u3088\u3046\u306b\u30aa\u30a4\u30e9\u30fc\u306f\u3001\u7d20\u6570\u3067\u69cb\u6210\u3055\u308c\u308b\u7121\u9650\u7a4d\u3068<span><a href=\"https:\/\/science-hub.click\/?p=69721\">\u5186<\/a><\/span>\u306e<span><a href=\"https:\/\/science-hub.click\/?p=48998\">\u8868\u9762\u7a4d<\/a><\/span>\u3068\u306e\u9593\u306b\u5947\u5999\u306a\u95a2\u4fc2\u3092\u78ba\u7acb\u3057\u307e\u3059\u3002\u95a2\u9023\u3059\u308b\u7d1a\u6570\u3092\u5408\u8a08\u3059\u308b\u554f\u984c\u306f\u3001\u30e1\u30f3\u30b4\u30ea\u554f\u984c\u3068\u3057\u3066\u9577\u3044\u9593\u77e5\u3089\u308c\u3066\u3044\u307e\u3057\u305f\u3002 1735\u5e74\u306b\u30aa\u30a4\u30e9\u30fc\u306b\u3088\u3063\u3066\u89e3\u6c7a\u3055\u308c\u307e\u3057\u305f<\/p><div align=\"left\"><div title=\"[\u62e1\u5927\u3059\u308b]\"><div align=\"left\"><p><i>f<\/i>\u306e\u30d5\u30fc\u30ea\u30a8\u5909\u63db<sub>\u306e<\/sub>\u4fc2\u6570 ( <i>cn<\/i> ) \u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002\u5947\u6570\u3067\u3042\u308b\u305f\u3081\u3001<span><a href=\"https:\/\/science-hub.click\/?p=99105\">\u4fc2\u6570<\/a><\/span><i>c<\/i> <sub>0<\/sub>\u306f\u30bc\u30ed\u3067\u3059\u3002<\/p><p> <i>c<\/i> <sub>n<\/sub>\u306e\u8a08\u7b97\u306f\u3001\u90e8\u5206\u3054\u3068\u306e\u7a4d\u5206\u3092\u4f7f\u7528\u3057\u3066\u6b21\u306e\u3088\u3046\u306b\u884c\u308f\u308c\u307e\u3059\u3002 <\/p><center><div class=\"math-formual notranslate\">$$ {c_n = \\frac 1{\\sqrt {2\\pi}}\\int_{-\\pi}^{\\pi} t \\ e^{(-int)} dt =  \\frac 1{\\sqrt {2\\pi}}\\left[ \\frac in t\\ e^{(-int)}\\right]_{-\\pi}^{\\pi} &#8211; \\frac 1{\\sqrt {2\\pi}}\\int_{-\\pi}^{\\pi} \\frac in e^{(-int)} dt = (-1)^n\\ \\frac {i\\sqrt {2\\pi}}n } $$<\/div><\/center><p> Parseval \u306e\u7b49\u4fa1\u6027\u306b\u3088\u308a\u3001\u6b21\u306e\u3053\u3068\u304c\u8a3c\u660e\u3055\u308c\u307e\u3059\u3002 <\/p><center><div class=\"math-formual notranslate\">$$ {\\sum_{n \\in \\mathbb Z} c_n \\bar {c_n} = 4\\pi \\sum_{n=1}^{+\\infty} \\frac1{n^2} = \\int_{-\\pi}^{\\pi} t^2dt = \\frac {2\\pi^3}3 } $$<\/div><\/center><\/div><\/div><\/div><\/div><h2 class=\"ref_link\">\u53c2\u8003\u8cc7\u6599<\/h2><ol><li><a class=\"notranslate\" href=\"https:\/\/ar.wikipedia.org\/wiki\/%D8%AC%D8%AF%D8%A7%D8%A1_%D8%A3%D9%88%D9%8A%D9%84%D8%B1\">\u062c\u062f\u0627\u0621 \u0623\u0648\u064a\u0644\u0631 \u2013 arabe<\/a><\/li><li> <a class=\"notranslate\" href=\"https:\/\/ca.wikipedia.org\/wiki\/Producte_d%27Euler\">Producte d&#8217;Euler \u2013 catalan<\/a><\/li><li> <a class=\"notranslate\" href=\"https:\/\/de.wikipedia.org\/wiki\/Euler-Produkt\">Euler-Produkt \u2013 allemand<\/a><\/li><li> <a class=\"notranslate\" href=\"https:\/\/en.wikipedia.org\/wiki\/Euler_product\">Euler product \u2013 anglais<\/a><\/li><li> <a class=\"notranslate\" href=\"https:\/\/es.wikipedia.org\/wiki\/Producto_de_Euler\">Producto de Euler \u2013 espagnol<\/a><\/li><li> <a class=\"notranslate\" href=\"https:\/\/fa.wikipedia.org\/wiki\/%D8%B6%D8%B1%D8%A8_%D8%A7%D9%88%DB%8C%D9%84%D8%B1\">\u0636\u0631\u0628 \u0627\u0648\u06cc\u0644\u0631 \u2013 persan<\/a><\/li><\/ol><\/div>\n<div class=\"feature-video\">\n <h2>\n  \u30aa\u30a4\u30e9\u30fc\u7a4d\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\u30fb\u95a2\u9023\u52d5\u753b\n <\/h2>\n <div class=\"video-item\">\n  \n  <figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\">\n   <div class=\"wp-block-embed__wrapper\">\n    <iframe loading=\"lazy\" title=\"\u3010\u6570\u5b66\u2162\u306e\u307f\u3011\u30e9\u30de\u30cc\u30b8\u30e3\u30f3\u306e2\u6b21\u306e\u30aa\u30a4\u30e9\u30fc\u7a4d\u306e\u5c0e\u51fa\u3010\u9b3c\u624d\u3011\u3010\u30a4\u30f3\u30c9\u306e\u9b54\u8853\u5e2b\u3011\u3010\u5929\u624d\u7684\u767a\u898b\u3011\/ How to derive the Ramanujan&#039;s quadratic Euler product\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/azRM6nzgzcM?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n   <\/div>\n  <\/figure>\n  \n <\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u5c0e\u5165 \u30ec\u30aa\u30f3\u30cf\u30eb\u30c8\u30fb\u30aa\u30a4\u30e9\u30fc \u6570\u5b66\u3001\u3088\u308a\u6b63\u78ba\u306b\u306f\u89e3\u6790\u7684\u6574\u6570\u8ad6\u3067\u306f\u3001\u30aa\u30a4\u30e9\u30fc\u7a4d\u306f\u3001\u7d20\u6570\u306b\u3088\u3063\u3066\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u4ed8\u3051\u3055\u308c\u305f\u7121\u9650\u7a4d\u3078\u306e\u62e1\u5f35\u3067\u3059\u3002 \u3053\u308c\u306b\u3088\u308a\u7d20\u6570\u306e\u5206\u5e03\u306e\u6e2c\u5b9a\u304c\u53ef\u80fd\u306b\u306a\u308a\u3001 \u30ea\u30fc\u30de\u30f3\u306e\u30bc\u30fc\u30bf\u95a2\u6570\u3068\u5bc6\u63a5\u306b\u95a2\u9023\u3057\u3066\u3044\u307e\u3059 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":92600,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"https:\/\/img.youtube.com\/vi\/GmRG7_WimmE\/0.jpg","fifu_image_alt":"\u30aa\u30a4\u30e9\u30fc\u7a4d\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac","footnotes":""},"categories":[5],"tags":[81911,11,13,81910,10,14,12,8,16,2386,15,9],"class_list":["post-92599","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-dictionary","tag-chateau-depoisses","tag-techniques","tag-technologie","tag-digital-s","tag-actualite","tag-news","tag-dossier","tag-definition","tag-sciences","tag-produit","tag-article","tag-explications"],"_links":{"self":[{"href":"https:\/\/science-hub.click\/index.php?rest_route=\/wp\/v2\/posts\/92599"}],"collection":[{"href":"https:\/\/science-hub.click\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/science-hub.click\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/science-hub.click\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/science-hub.click\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=92599"}],"version-history":[{"count":0,"href":"https:\/\/science-hub.click\/index.php?rest_route=\/wp\/v2\/posts\/92599\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/science-hub.click\/index.php?rest_route=\/wp\/v2\/media\/92600"}],"wp:attachment":[{"href":"https:\/\/science-hub.click\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=92599"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/science-hub.click\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=92599"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/science-hub.click\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=92599"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}