[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki5\/archives\/8864#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki5\/archives\/8864","headline":"\u7b97\u8853\u7d1a\u6570\u5b9a\u7406 – Wikipedia","name":"\u7b97\u8853\u7d1a\u6570\u5b9a\u7406 – Wikipedia","description":"\u3053\u306e\u8a18\u4e8b\u306f\u691c\u8a3c\u53ef\u80fd\u306a\u53c2\u8003\u6587\u732e\u3084\u51fa\u5178\u304c\u5168\u304f\u793a\u3055\u308c\u3066\u3044\u306a\u3044\u304b\u3001\u4e0d\u5341\u5206\u3067\u3059\u3002\u51fa\u5178\u3092\u8ffd\u52a0\u3057\u3066\u8a18\u4e8b\u306e\u4fe1\u983c\u6027\u5411\u4e0a\u306b\u3054\u5354\u529b\u304f\u3060\u3055\u3044\u3002\u51fa\u5178\u691c\u7d22?:\u00a0“\u7b97\u8853\u7d1a\u6570\u5b9a\u7406”\u00a0\u2013\u00a0\u30cb\u30e5\u30fc\u30b9\u00a0\u00b7 \u66f8\u7c4d\u00a0\u00b7 \u30b9\u30ab\u30e9\u30fc\u00a0\u00b7 CiNii\u00a0\u00b7 J-STAGE\u00a0\u00b7 NDL\u00a0\u00b7 dlib.jp\u00a0\u00b7 \u30b8\u30e3\u30d1\u30f3\u30b5\u30fc\u30c1\u00a0\u00b7 TWL\uff082015\u5e7411\u6708\uff09 \u7b97\u8853\u7d1a\u6570\u5b9a\u7406\uff08\u3055\u3093\u3058\u3085\u3064\u304d\u3085\u3046\u3059\u3046\u3066\u3044\u308a\u3001theorem on arithmetic progressions\uff09\u306f\u3001\u521d\u9805\u3068\u516c\u5dee\u304c\u4e92\u3044\u306b\u7d20\u3067\u3042\u308b\u7b97\u8853\u7d1a\u6570(\u7b49\u5dee\u6570\u5217)\u306b\u306f\u7121\u9650\u306b\u7d20\u6570\u304c\u5b58\u5728\u3059\u308b\u3001\u3068\u3044\u3046\u5b9a\u7406\u3067\u3042\u308b\u3002\u30da\u30fc\u30bf\u30fc\u30fb\u30b0\u30b9\u30bf\u30d5\u30fb\u30c7\u30a3\u30ea\u30af\u30ec\u304c1837\u5e74\u306b\u30c7\u30a3\u30ea\u30af\u30ec\u306eL\u95a2\u6570\u3092\u7528\u3044\u3066\u521d\u3081\u3066\u8a3c\u660e\u3057\u305f\u3002\u305d\u306e\u305f\u3081\u3001\u5b9a\u7406\u306f\u3057\u3070\u3057\u3070\u30c7\u30a3\u30ea\u30af\u30ec\u306e\u7b97\u8853\u7d1a\u6570\u5b9a\u7406\u3068\u547c\u3070\u308c\u308b\u3002 \u5b9a\u7406\u306e\u8a00\u3044\u63db\u3048\u3068\u3057\u3066\u3001 gcd(a,b)=1{displaystyle gcd(a,b)=1} \u3067\u3042\u308b\u81ea\u7136\u6570 a, b \u306b\u5bfe\u3057\u3001","datePublished":"2022-09-11","dateModified":"2022-09-11","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki5\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki5\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/6\/64\/Question_book-4.svg\/50px-Question_book-4.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/6\/64\/Question_book-4.svg\/50px-Question_book-4.svg.png","height":"39","width":"50"},"url":"https:\/\/wiki.edu.vn\/jp\/wiki5\/archives\/8864","about":["Wiki"],"wordCount":17952,"articleBody":"\u3053\u306e\u8a18\u4e8b\u306f\u691c\u8a3c\u53ef\u80fd\u306a\u53c2\u8003\u6587\u732e\u3084\u51fa\u5178\u304c\u5168\u304f\u793a\u3055\u308c\u3066\u3044\u306a\u3044\u304b\u3001\u4e0d\u5341\u5206\u3067\u3059\u3002\u51fa\u5178\u3092\u8ffd\u52a0\u3057\u3066\u8a18\u4e8b\u306e\u4fe1\u983c\u6027\u5411\u4e0a\u306b\u3054\u5354\u529b\u304f\u3060\u3055\u3044\u3002\u51fa\u5178\u691c\u7d22?:\u00a0“\u7b97\u8853\u7d1a\u6570\u5b9a\u7406”\u00a0\u2013\u00a0\u30cb\u30e5\u30fc\u30b9\u00a0\u00b7 \u66f8\u7c4d\u00a0\u00b7 \u30b9\u30ab\u30e9\u30fc\u00a0\u00b7 CiNii\u00a0\u00b7 J-STAGE\u00a0\u00b7 NDL\u00a0\u00b7 dlib.jp\u00a0\u00b7 \u30b8\u30e3\u30d1\u30f3\u30b5\u30fc\u30c1\u00a0\u00b7 TWL\uff082015\u5e7411\u6708\uff09\u7b97\u8853\u7d1a\u6570\u5b9a\u7406\uff08\u3055\u3093\u3058\u3085\u3064\u304d\u3085\u3046\u3059\u3046\u3066\u3044\u308a\u3001theorem on arithmetic progressions\uff09\u306f\u3001\u521d\u9805\u3068\u516c\u5dee\u304c\u4e92\u3044\u306b\u7d20\u3067\u3042\u308b\u7b97\u8853\u7d1a\u6570(\u7b49\u5dee\u6570\u5217)\u306b\u306f\u7121\u9650\u306b\u7d20\u6570\u304c\u5b58\u5728\u3059\u308b\u3001\u3068\u3044\u3046\u5b9a\u7406\u3067\u3042\u308b\u3002\u30da\u30fc\u30bf\u30fc\u30fb\u30b0\u30b9\u30bf\u30d5\u30fb\u30c7\u30a3\u30ea\u30af\u30ec\u304c1837\u5e74\u306b\u30c7\u30a3\u30ea\u30af\u30ec\u306eL\u95a2\u6570\u3092\u7528\u3044\u3066\u521d\u3081\u3066\u8a3c\u660e\u3057\u305f\u3002\u305d\u306e\u305f\u3081\u3001\u5b9a\u7406\u306f\u3057\u3070\u3057\u3070\u30c7\u30a3\u30ea\u30af\u30ec\u306e\u7b97\u8853\u7d1a\u6570\u5b9a\u7406\u3068\u547c\u3070\u308c\u308b\u3002  \u5b9a\u7406\u306e\u8a00\u3044\u63db\u3048\u3068\u3057\u3066\u3001gcd(a,b)=1{displaystyle gcd(a,b)=1} \u3067\u3042\u308b\u81ea\u7136\u6570 a, b \u306b\u5bfe\u3057\u3001an+b{displaystyle an+b} (n \u306f\u81ea\u7136\u6570)\u3068\u66f8\u3051\u308b\u7d20\u6570\u304c\u7121\u9650\u306b\u5b58\u5728\u3059\u308b\u3001\u3068\u3057\u3066\u3082\u3088\u3044\u3002\u3055\u3089\u306b\u3001\u305d\u306e\u3088\u3046\u306a\u7d20\u6570\u306e\u9006\u6570\u548c\u306f\u767a\u6563\u3057\u3001 x\u4ee5\u4e0b\u306e\u8a72\u5f53\u3059\u308b\u7d20\u6570\u306e\u9006\u6570\u306e\u548c\u306f  \u223c(log\u2061log\u2061x)\/\u03c6(a){displaystyle sim (log log x)\/varphi (a)}\u3092\u6e80\u305f\u3059\u3002\u3053\u306e\u5b9a\u7406\u306f\u30ac\u30a6\u30b9\u304c\u4e88\u60f3\u3057\u305f\u3068\u3055\u308c\u308b\u304c\u3001\u8a3c\u660e\u306f1837\u5e74\u306b\u30c7\u30a3\u30ea\u30af\u30ec\u304cL\u95a2\u6570\u3092\u5c0e\u5165\u3057\u3066\u884c\u3063\u305f\u3002\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306b\u3088\u308b\u7d20\u6570\u304c\u7121\u9650\u306b\u5b58\u5728\u3059\u308b\u3068\u3044\u3046\u5b9a\u7406\u3092\u8d8a\u3048\u3066\u3001\u8fd1\u4ee3\u306e\u6570\u5b66\u304c\u5927\u304d\u304f\u9032\u6b69\u3057\u305f\u3053\u3068\u3092\u793a\u3057\u305f\u3002Table of Contents\u7b97\u8853\u7d1a\u6570\u306e\u7d20\u6570\u5b9a\u7406[\u7de8\u96c6]\u8a18\u53f7[\u7de8\u96c6]\u30c7\u30a3\u30ea\u30af\u30ec\u6307\u6a19[\u7de8\u96c6]\u30c7\u30a3\u30ea\u30af\u30ec\u7d1a\u6570[\u7de8\u96c6]\u30c7\u30a3\u30ea\u30af\u30ec\u306e\u30a8\u30eb\u95a2\u6570[\u7de8\u96c6]\u88dc\u984c[\u7de8\u96c6]\u7b97\u8853\u7d1a\u6570\u5b9a\u7406\u306e\u8a3c\u660e[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u7b97\u8853\u7d1a\u6570\u306e\u7d20\u6570\u5b9a\u7406[\u7de8\u96c6]\u516c\u5dee\u304c a \u3067\u3042\u308b\u7b49\u5dee\u6570\u5217\u306f\u521d\u9805\u3092 1 \u304b\u3089   a\u22121{displaystyle a-1} \u306e\u9593\u306b\u53d6\u308b\u3068\u304d\u305d\u306e\u521d\u9805\u304c a \u3068\u4e92\u3044\u306b\u7d20\u3067\u3042\u308b\u3082\u306e\u304c \u03c6(a){displaystyle varphi (a)} \u901a\u308a\u3042\u308b\u3002\u3053\u3053\u3067 \u03c6(a){displaystyle varphi (a)} \u306f\u30aa\u30a4\u30e9\u30fc\u306e\u03c6\u95a2\u6570\u3067\u3042\u308b\u3002\u3053\u308c\u3089 \u03c6(a){displaystyle varphi (a)} \u500b\u306e\u7b49\u5dee\u6570\u5217\u306b\u7d20\u6570\u306f\u305d\u308c\u305e\u308c\u307b\u307c\u5747\u7b49\u306b\u5206\u5e03\u3057\u3066\u3044\u308b\u3002\u7d20\u6570\u5b9a\u7406\u306e\u62e1\u5f35\u3068\u3057\u3066\u3001\u6b21\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002\u521d\u9805 b \u3068\u516c\u5dee a \u304c\u4e92\u3044\u306b\u7d20\u3067\u3042\u308b\u7b49\u5dee\u6570\u5217\u306b\u542b\u307e\u308c\u308b\u7d20\u6570\u3067\u3001x \u4ee5\u4e0b\u306e\u3082\u306e\u306e\u6570\u3092 \u03c0a,b(x){displaystyle pi _{a,b}(x)} \u3067\u8868\u3059\u3068\u304d\u3001\u03c0a,b(x)\u223c1\u03c6(a)Li(x){displaystyle pi _{a,b}(x)sim {frac {1}{varphi (a)}}mathrm {Li} (x)}\u30c7\u30a3\u30ea\u30af\u30ec\u304c\u7b97\u8853\u7d1a\u6570\u5b9a\u7406\u3092\u8a3c\u660e\u3057\u305f\u5f53\u6642\u3001\u7d20\u6570\u5b9a\u7406\u3082\u307e\u3060\u8a3c\u660e\u3055\u308c\u3066\u3044\u306a\u304b\u3063\u305f\u305f\u3081\u3053\u306e\u5f62\u306f\u4e88\u60f3\u306b\u904e\u304e\u306a\u304b\u3063\u305f\u304c\u3001\u5f8c\u306b\u7d20\u6570\u5b9a\u7406\u3068\u540c\u69d8\u306b\u30b7\u30e3\u30eb\u30eb\uff1d\u30b8\u30e3\u30f3\u30fb\u30c9\u30fb\u30e9\u30fb\u30f4\u30a1\u30ec\u30fc\u30fb\u30d7\u30fc\u30b5\u30f3\uff08\u30d5\u30e9\u30f3\u30b9\u8a9e\u7248\uff09\u306b\u3088\u3063\u3066\u8a3c\u660e\u3055\u308c\u305f\u3002\u3053\u306e\u5b9a\u7406\u3092\u7b97\u8853\u7d1a\u6570\u306e\u7d20\u6570\u5b9a\u7406\u3068\u547c\u3076\u3002\u7d20\u6570\u304c\u7121\u6570\u306b\u5b58\u5728\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u306f\u53e4\u4ee3\u304b\u3089\u77e5\u3089\u308c\u3066\u304d\u305f\u4e8b\u5b9f\u3067\u3042\u308b\u304c\u3001\u30bc\u30fc\u30bf\u95a2\u6570\u306e\u30aa\u30a4\u30e9\u30fc\u4e57\u7a4d\u8868\u793a\u306b\u3082\u7aef\u7684\u306b\u9855\u308f\u308c\u3066\u3044\u308b\u3002\u03b6(s)=\u2211n=1\u221e1ns=\u220fp11\u2212p\u2212s{displaystyle zeta (s)=sum _{n=1}^{infty }{frac {1}{n^{s}}}=prod _{p}{frac {1}{1-p^{-s}}}}\u3053\u306e\u5de6\u8fba\u306e\u30bc\u30fc\u30bf\u95a2\u6570\u306fs=1{displaystyle s=1}\u306b\u6975\u3092\u6301\u3064\u304b\u3089\u3001\u53f3\u8fba\u3082\u767a\u6563\u3057\u306a\u3051\u308c\u3070\u306a\u3089\u305a\u3001\u305d\u306e\u305f\u3081\u306b\u306f\u7121\u9650\u500b\u306e\u7d20\u6570\u304c\u5b58\u5728\u3057\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u3002\u3053\u308c\u306b\u5023\u3044\u3001\u4efb\u610f\u306e\u7b97\u8853\u7d1a\u6570\u306b\u542b\u307e\u308c\u308b\u7d20\u6570\u3067\u69cb\u6210\u3055\u308c\u305f\u7dcf\u548c\u304c\u767a\u6563\u3059\u308b\u3053\u3068\u3092\u3082\u3063\u3066\u30c7\u30a3\u30ea\u30af\u30ec\u306e\u7b97\u8853\u7d1a\u6570\u5b9a\u7406\u304c\u8a3c\u660e\u3055\u308c\u308b\u3002\u8a18\u53f7[\u7de8\u96c6]\u4ee5\u4e0b\u306e\u8a18\u53f7\u3092\u7528\u3044\u308b\u3002\u30c7\u30a3\u30ea\u30af\u30ec\u6307\u6a19[\u7de8\u96c6]\u6574\u6570\u304b\u3089\u8907\u7d20\u6570\u3078\u306e\u5199\u50cf\u03c7:Z\u21a6C{displaystyle chi :mathbb {Z} mapsto mathbb {C} }\u3067\u4e0b\u8a18\u306e\u6027\u8cea\u3092\u6e80\u305f\u3059\u3082\u306e\u3092\u6cd5d{displaystyle d}\u306e\u30c7\u30a3\u30ea\u30af\u30ec\u6307\u6a19\u3068\u3044\u3046\u3002(d,n)=1\u21d4\u03c7(n)\u22600{displaystyle (d,n)=1Leftrightarrow chi (n)neq 0}\u03c7(n1)\u03c7(n2)=\u03c7(n1n2){displaystyle chi (n_{1})chi (n_{2})=chi (n_{1}n_{2})}\u03c7(n+d)=\u03c7(n){displaystyle chi (n+d)=chi (n)}\u7279\u306b\u3001\u03c70(n)\u22600{displaystyle chi _{0}(n)neq 0}\u306a\u3089\u3070\u03c70(n)=1{displaystyle chi _{0}(n)=1}\u3068\u306a\u308b\u03c70(n){displaystyle chi _{0}(n)}\u3092\u81ea\u660e\u306a\u6307\u6a19\u3068\u547c\u3076\u3002\u6b63\u306e\u6574\u6570d{displaystyle d}\u306b\u3064\u304d\u03c6(d){displaystyle varphi (d)}\u500b\u306e\u30c7\u30a3\u30ea\u30af\u30ec\u6307\u6a19\u304c\u3042\u308a\u3001\u305d\u308c\u3089\u306f\u7fa4\u3092\u6210\u3059\u3002\u30c7\u30a3\u30ea\u30af\u30ec\u6307\u6a19\u306b\u306f\u76f4\u4ea4\u6027\u304c\u3042\u308b\u3002\u2211n=1d\u03c7(n)={\u03c6(d)\u03c7=\u03c700\u03c7\u2260\u03c70{displaystyle sum _{n=1}^{d}chi (n)={begin{cases}varphi (d)&chi =chi _{0}&chi neq chi _{0}end{cases}}}\u2211\u03c7\u03c7(n)={\u03c6(d)n\u226110n\u22621{displaystyle sum _{chi }chi (n)={begin{cases}varphi (d)&nequiv 1&nnot equiv 1end{cases}}}\u30c7\u30a3\u30ea\u30af\u30ec\u7d1a\u6570[\u7de8\u96c6]\u6b21\u5f0f\u306e\u5f62\u306e\u7d1a\u6570\u3092\u30c7\u30a3\u30ea\u30af\u30ec\u7d1a\u6570\u3068\u3044\u3046\u3002\u2211n=1\u221eanns{displaystyle sum _{n=1}^{infty }{frac {a_{n}}{n^{s}}}}\u30c7\u30a3\u30ea\u30af\u30ec\u7d1a\u6570\u306f\u3001|\u2211n=1\u221eanns|\u2264sup|an|\u2211n=1\u221e1n\u211cs\u2264sup|an|(1+\u222bu=1\u221eduu\u211cs){displaystyle left|sum _{n=1}^{infty }{frac {a_{n}}{n^{s}}}right|leq sup {|a_{n}|}sum _{n=1}^{infty }{frac {1}{n^{Re {s}}}}leq sup {|a_{n}|}left(1+int _{u=1}^{infty }{frac {du}{u^{Re {s}}}}right)}\u3067\u3042\u308b\u304b\u3089\u3001an{displaystyle a_{n}}\u304c\u6709\u754c\u3067\u3042\u308c\u3070"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki5\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki5\/archives\/8864#breadcrumbitem","name":"\u7b97\u8853\u7d1a\u6570\u5b9a\u7406 – Wikipedia"}}]}]