[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/21152#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/21152","headline":"Ramanujansumme  – \u30a6\u30a3\u30ad\u30da\u30c7\u30a3\u30a2","name":"Ramanujansumme  – \u30a6\u30a3\u30ad\u30da\u30c7\u30a3\u30a2","description":"before-content-x4 \u3044\u3064 \u30e9\u30de\u30cc\u30b8\u30e3\u30f3\u30b5\u30e0 \u6570\u7406\u8ad6\u306e\u7279\u5b9a\u306e\u6709\u9650\u5408\u8a08\u3001\u6570\u5b66\u306e\u30b5\u30d6\u30a8\u30ea\u30a2 c q\uff08 n \uff09\uff09 {displaystyle c_ {q}\uff08n\uff09} after-content-x4 \u81ea\u7136\u6570\u306e\u4fa1\u5024 Q {displaystyle q} \u305d\u3057\u3066\u6574\u6570 n {displaystyle n}","datePublished":"2021-05-07","dateModified":"2021-05-07","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?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\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6b62363d969b768868a3f2c5052face3e255906f","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6b62363d969b768868a3f2c5052face3e255906f","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/21152","wordCount":12836,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4\u3044\u3064 \u30e9\u30de\u30cc\u30b8\u30e3\u30f3\u30b5\u30e0 \u6570\u7406\u8ad6\u306e\u7279\u5b9a\u306e\u6709\u9650\u5408\u8a08\u3001\u6570\u5b66\u306e\u30b5\u30d6\u30a8\u30ea\u30a2 c q\uff08 n \uff09\uff09 {displaystyle c_ {q}\uff08n\uff09}        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u81ea\u7136\u6570\u306e\u4fa1\u5024 Q {displaystyle q} \u305d\u3057\u3066\u6574\u6570 n {displaystyle n} \u53c2\u7167\u3055\u308c\u307e\u3059\u3002\u5f7c\u5973\u306f\u901a\u308a\u629c\u3051\u3066\u3044\u307e\u3059        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4cq\uff08 n \uff09\uff09 = \u2211a=1(a,q)=1qe2\u03c0iaqn{displaystyle c_ {q}\uff08n\uff09= sum _ {a = 1 atop\uff08a\u3001q\uff09= 1}^{q} e^{2pi i {frac {a} {q}}}}}} \u5b9a\u7fa9\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u30b9\u30da\u30eb \uff08 a \u3001 Q \uff09\uff09 {displaystyle\uff08a\u3001q\uff09} \u306e\u6700\u5927\u306e\u5171\u6709\u5206\u5272\u8005\u3092\u8868\u3057\u3066\u3044\u307e\u3059 a {displaystyle a} \u3068        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Q {displaystyle q} \u3001\u5408\u8a08\u306f\u6570\u5b57\u306e\u4e0a\u306b\u5e83\u304c\u3063\u3066\u3044\u307e\u3059 a {displaystyle a} \u3068 \u521d\u3081 \u2264 a \u2264 Q {displaysStyle 1LQ leq} 1\u3064\u3082 Q {displaystyle q} \u53c2\u52a0\u3059\u308b\u5916\u56fd\u4eba\u3067\u3059\u3002\u500b\u3005\u306e\u30b5\u30de\u30f3\u30c9\u306f\u3001\u3057\u3063\u304b\u308a\u3057\u305f\u8907\u96d1\u306a\u30e6\u30cb\u30c3\u30c8\u30eb\u30fc\u30c8\u306e\u5f37\u529b\u3067\u3059\u3002 S.\u30e9\u30de\u30cc\u30b8\u30e3\u30f3\u306f1916\u5e74\u306b\u3053\u308c\u3089\u306e\u91d1\u984d\u3092\u5c0e\u5165\u3057\u307e\u3057\u305f\u3002 [\u521d\u3081] \u3042\u306a\u305f\u306f\u91cd\u8981\u306a\u5f79\u5272\u3092\u679c\u305f\u3057\u307e\u3059 \u5186\u5f62\u306e\u65b9\u6cd5 Hardy\u3001Littlewood\u3001Winogradow\u306e\u5f8c\u3002 [2] \u2192\u4e09\u89d2\u591a\u9805\u5f0f\u3082\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002 Ramanujansumme\u306f\u3001\u3053\u308c\u3089\u306e\u6a5f\u80fd\u306e\u5206\u6790\u7684\u7d99\u7d9a\u3092\u53ef\u80fd\u306b\u3059\u308b\u6570\u306e\u7406\u8ad6\u7684\u6a5f\u80fd\u306e\u8208\u5473\u6df1\u3044\u8868\u73fe\u3092\u7372\u5f97\u3067\u304d\u307e\u3059\u3002 \u660e\u78ba\u306a\u30d7\u30ec\u30bc\u30f3\u30c6\u30fc\u30b7\u30e7\u30f3\u306e\u305f\u3081\u306b\u3001\u6570\u306e\u7406\u8ad6\u306f\u7565\u3055\u308c\u307e\u3059 \u305d\u3046\u3067\u3059 \uff08 \u30d0\u30c4 \uff09\uff09 = \u305d\u3046\u3067\u3059 2\u03c0ix{displaystyle e\uff08x\uff09= e^{2pi ix}} \u66f8\u304b\u308c\u305f\u3082\u306e\u3068\u95a2\u6570 \u305d\u3046\u3067\u3059 {displaystyle e} \u610f\u5fd7\u3068\u3057\u3066 \u591a\u6570\u306e\u7406\u8ad6\u7684\u6307\u6570\u95a2\u6570 \u5c02\u7528\u3002 [3] \u6570\u306e\u7406\u8ad6\u7684\u6307\u6570\u95a2\u6570\u3067\u306f\u3001\u30e9\u30de\u30cc\u30b8\u30e3\u30ca\u30f3\u306f c q\uff08 n \uff09\uff09 {displaystyle c_ {q}\uff08n\uff09} \u3044\u3064 cq\uff08 n \uff09\uff09 = \u2211a=1(a,q)=1q\u305d\u3046\u3067\u3059 (aq\u22c5n){displaystyle c_ {q}\uff08n\uff09= sum _ {a = 1 atop\uff08a\u3001q\uff09= 1}^{q} eleft\uff08{frac {a} {q}} cdot norright\uff09}} \u66f8\u304f\u3002 \u5b8c\u5168\u306a\u6570\u5b57\u306e\u5834\u5408 a {displaystyle a} \u3068 b {displaystyle b} \u3042\u306a\u305f\u304c\u66f8\u304f a ‘ b {b}\u306e\u4e2d\u306edisplaystyle \u3001\u6574\u6570\u306e\u5834\u5408\u306f\u300ca shares b\u300d\u3092\u8aad\u3093\u3067\u304f\u3060\u3055\u3044 c {displaystyle c} \u3067\u5b58\u5728\u3057\u307e\u3057\u305f b = a de c {displaystyle b = acdot c} \u9069\u7528\u53ef\u80fd\u3067\u3059\u3002\u305d\u306e\u3088\u3046\u306a\u6570\u304c\u306a\u3044\u5834\u5408\u30011\u3064\u306f\u66f8\u3044\u3066\u3044\u307e\u3059 a \u2224 b {displaystyle anmid b} \u3001\u300ca\u306fb\u3092\u5171\u6709\u3057\u306a\u3044\u300d\u3068\u8aad\u307f\u307e\u3059\u3002\u5408\u8a08\u8a18\u53f7 \u2211d\u2223mf \uff08 d \uff09\uff09 {DisplayStyle Text Style Sum {D\u3001Mid\u3001M} f\uff08d\uff09} \u7dcf\u5408\u6307\u6570\u3092\u610f\u5473\u3057\u307e\u3059 d {displaystyle d} \u306e\u3059\u3079\u3066\u306e\u80af\u5b9a\u7684\u306a\u9664\u6570 m {displaystyle m} \u901a\u904e\u3057\u307e\u3059\u3002\u7d20\u6570\u306e\u52b9\u529b\u306e\u305f\u3081 p k\u3001 k \u2208 n \u2216 { 0 } {displaystyle p^{k}\u3001kin mathbb {n} setminus lbrace 0rbrace} \u305d\u3057\u3066\u6574\u6570 b {displaystyle b} \u3042\u306a\u305f\u304c\u66f8\u304f p k\u2225 b {displaystyle p^{k} parallel b} \uff08\u8aad\u3093\u3060\u300d p k{displaystyle p^{k}} B\u3092\u6b63\u78ba\u306b\u5171\u6709\u3059\u308b “\uff09\u3001if p k‘ b \u3001 {displaystyle p^{k} mid b\u3001} \u3057\u304b\u3057 p k+1\u2224 b {displaystyle p^{k+1} nmid b}  – \u3064\u307e\u308a\u3001if \uff08 p k+1\u3001 b \uff09\uff09 = p k{displaystyle\uff08p^{k+1}\u3001b\uff09= p^{k}} \u3002 \u5909\u6570\u306e1\u3064\u3092\u4fdd\u6301\u3057\u307e\u3059 Q {displaystyle q} \u307e\u305f n {displaystyle n} \u30e9\u30de\u30cc\u30b8\u30e3\u30ca\u30f3\u3067 c q\uff08 n \uff09\uff09 {displaystyle c_ {q}\uff08n\uff09} \u6700\u5f8c\u306b\u3001\u4ed6\u306e\u5909\u6570\u306b\u5fdc\u3058\u3066\u6570\u5b57\u306e\u7406\u8ad6\u7684\u95a2\u6570\u3092\u53d6\u5f97\u3057\u307e\u3059\u3002 n {displaystyle n} \u3053\u306e\u7528\u8a9e\u306e\u305f\u3081\u306b \u5909\u6570 \u306e\u4e0a n \u2208 n \u2216 { 0 } {displaystyle nin mathbb {n} setminus lbrace 0rbrace} \u5236\u9650\u3055\u308c\u307e\u3059\u3002\u4f1a\u793e\u3068 Q {displaystyle q} \u95a2\u6570\u3067\u3059 n \u21a6 c q\uff08 n \uff09\uff09 {displaystyle nmapsto c_ {q}\uff08n\uff09}  Q {displaystyle q}  – \u5468\u671f\u3001\u3064\u307e\u308a\u3001\u9069\u7528\u3055\u308c\u307e\u3059 cq\uff08 m \uff09\uff09 = cq\uff08 n \uff09\uff09 {displaystyle c_ {q}\uff08m\uff09= c_ {q}\uff08n\uff09} \u3001\u6edd m \u559c\u3093\u3067 n (modq)\u3001 n \u3001 m \u2208 Z{displaystyle mequiv n {pmod {q}} ,, n\u3001min mathbb {z}} \u3002 \u5408\u8a08\u306b\u5916\u56fd\u306e\u72b6\u614b\u3092\u96e2\u308c\u308b\u3068\u3001 \u2211a=1q\u305d\u3046\u3067\u3059 (aq\u22c5n)= {qfallsn\u22610(modq)0sonst,{displaystyle sum _ {a = 1}^{q} eleft\uff08{frac {a} {q}} cdot nright\uff09= {begin {cases} qquad {falls}}; nequiv 0 {pmod {q}} \\ 0 {sonst\u3001}}}}}}} \u306a\u305c\u306a\u3089\u3001\u5de6\u5074\u306f\u5e7e\u4f55\u5b66\u7684\u306a\u5408\u8a08\u3060\u304b\u3089\u3067\u3059\u3002\u306e\u6700\u5927\u306e\u5171\u6709\u5206\u5272\u8005\u306b\u5fdc\u3058\u3066\u5408\u8a08\u3067\u4e26\u3079\u66ff\u3048\u307e\u3059 Q {displaystyle q} \u3068 a {displaystyle a} \u3001\u6b21\u306b\u3001\u6570\u7406\u8ad6\u7684\u6a5f\u80fd\u306e\u76f4\u63a5\u6298\u308a\u305f\u305f\u307f\u304c\u3042\u308a\u307e\u3059 Q \u21a6 c q\uff08 n \uff09\uff09 {displaystyle qmapsto c_ {q}\uff08n\uff09} \u4e00\u5b9a\u306e\u95a2\u6570\u3067 \u79c1 0\uff08 Q \uff09\uff09 = \u521d\u3081 {displaystyle i^{0}\uff08q\uff09= 1} \uff1a \u2211a=1q\u305d\u3046\u3067\u3059 (aq\u22c5n)= \u2211d\u2223q\u2211a=1(a,q)=dq\u305d\u3046\u3067\u3059 (aq\u22c5n)= \u2211d\u2223qcq\/d\uff08 n \uff09\uff09 {displaystyle sum _ {a = 1}^{q} eleft\uff08{frac {a} {q}} cdot nright\uff09= sum _ {dmid q} sum _ {a = 1 atop\uff08a\u3001q\uff09= d}^{q} eleft\uff08{a} k} {a {a} k} summ {_ _ _ _ _ _ _ _ _ _ _ _ k {q\/d}\uff08n\uff09} \u3002 \u3053\u308c\u306b\u7d9a\u3044\u3066\u3001M\u00f6biissian\u53cd\u8ee2\u5f0f\u304c\u7d9a\u304d\u307e\u3059\u3002 cq\uff08 n \uff09\uff09 = \u2211d\u2223qn\u22610(modd)m (qd)de d = \u2211d\u2223(q,n)m (qd)de d \u3002 {displaystyle c_ {q}\uff08n\uff09= sum _ {{dmid q} atop {nequiv 0 {pmod {d}}}} mu left\uff08{frac {q} {d}} right\uff09cdot d = sum _ {dmid\uff08q\u3001rac dmid\uff08q\u3001qut} {d\u3002 \u6b21\u306b\u3001\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002 out (q,r)=1{displaystyle\uff08q\u3001r\uff09= 1} \u7d9a\u304d\u307e\u3059 cqr(n)=cq(n)\u22c5cr(n){displaystyle c_ {qr}\uff08n\uff09= c_ {q}\uff08n\uff09cdot c_ {r}\uff08n\uff09} \u305d\u3057\u3066\u3001\u305d\u308c\u306f\u5e38\u306b\u9069\u7528\u3055\u308c\u307e\u3059 cq\uff08 n \uff09\uff09 = cq\uff08 \uff08 Q \u3001 n \uff09\uff09 \uff09\uff09 {displaystyle c_ {q}\uff08n\uff09= c_ {q}\uff08\uff08q\u3001n\uff09\uff09} \u3002 eulersche\u03c6\u95a2\u6570\u3092\u4ecb\u3057\u3066\u30e9\u30de\u30cc\u30b8\u30e3\u30f3\u306e\u91cf\u3092\u4f7f\u7528\u3067\u304d\u307e\u3059 \u30d5\u30a1\u30a4 {displaystyle varphi} \u304a\u3088\u3073\u30e1\u30d3\u30a6\u30b9\u95a2\u6570 m {displaystyle mu} \u4ee3\u8868\u3059\u308b\uff1a [4] cq(n)=\u03c6(q)\u22c5\u03bc(q(q,n))\u03c6(q(q,n)){displaystyle c_ {q}\uff08n\uff09= varphi\uff08q\uff09cdot {frac {mu left\uff08{frac {q} {\uff08q\u3001n\uff09}}\u53f3\uff09} {varphi\uff08{frac {q} {\uff08q\u3001n\uff09}}}}}}}}}}} \uff08\u305f\u3081\u306b n=0{displaystyle n = 0} \u3042\u306a\u305f\u304c\u8a2d\u5b9a\u3057\u305f (q,0)=q{displaystyle\uff08q\u30010\uff09= q} \u3001\u3088\u308a\u4e00\u822c\u7684\u3067\u3059 (q,n){displaystyle\uff08q\u3001n\uff09} \u3044\u3064 \u30dd\u30b8\u30c6\u30a3\u30d6 GGT\u30d5\u30a7\u30b9\u30c8\uff09\u3001 \u3042\u306a\u305f\u306e\u4fa1\u5024\u306f\u56fa\u5b9a\u3055\u308c\u3066\u3044\u307e\u3059 Q {displaystyle q} \u91d1\u984d\u306e\u89b3\u70b9\u304b\u3089 \u30d5\u30a1\u30a4 \uff08 Q \uff09\uff09 {displaystyle varphi\uff08q\uff09} \u9650\u5b9a\u3001 \u306f Q \/\uff08 Q \u3001 n \uff09\uff09 {displaystyle q\/\uff08q\u3001n\uff09} \u6b63\u65b9\u5f62\u3067\u306f\u306a\u3044\u306e\u3067\u3001\u305d\u3046\u3067\u3059 cq\uff08 n \uff09\uff09 = 0 {displaystyle c_ {q}\uff08n\uff09= 0} \u3002 \u6570\u5b57\u7406\u8ad6\u95a2\u6570\u3092\u8868\u793a\u3059\u308b\u30e9\u30de\u30cc\u30b8\u30e3\u30f3\u30b5\u30e0 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30e9\u30de\u30cc\u30b8\u30e3\u30f3\u306f\u3001\u3044\u304f\u3064\u304b\u306e\u91cd\u8981\u306a\u7279\u5225\u306a\u30b1\u30fc\u30b9\u306e\u305f\u3081\u306b\u3001\u3042\u306a\u305f\u304c\u3042\u306a\u305f\u306e\u5408\u8a08\u3067\u6570\u306e\u7406\u8ad6\u7684\u6a5f\u80fd\u306e\u8208\u5473\u6df1\u3044\u8868\u73fe\u3092\u7372\u5f97\u3067\u304d\u308b\u3053\u3068\u3092\u3059\u3067\u306b\u793a\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u76ee\u7684\u306e\u305f\u3081\u306b\u3001\u6700\u5927\u306e\u5171\u901a\u4ed5\u5206\u3051\u306e\u6570\u306e\u7406\u8ad6\u7684\u6a5f\u80fd\u306b\u5bfe\u3057\u3066\u3001\u7279\u5225\u306a\u30bf\u30a4\u30d7\u306e\u63a7\u3048\u3081\u306a\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u304c\u5c0e\u5165\u3055\u308c\u307e\u3059\u3002 [5] \u306a\u308c n \u2208 \u3068 \u3001 Q \u2208 n {displaystyle nin mathbb {z}\u3001qin mathbb {n}}} \u3068 f \uff1a n \u2192 c {displaystyle fcolon mathbb {n} to mathbb {c}} \u6570\u306e\u7406\u8ad6\u7684\u95a2\u6570\u3002\u6b21\u306b\u610f\u5473\u3057\u307e\u3059 Ff\uff08 n \u3001 Q \uff09\uff09 = \u2211a=1qf \uff08 \uff08 a \u3001 Q \uff09\uff09 \uff09\uff09 de \u305d\u3046\u3067\u3059 (\u2212aq\u22c5n){displaystyle f_ {f}\uff08n\u3001q\uff09= sum _ {a = 1}^{q} f\uff08\uff08a\u3001q\uff09\uff09cdot eleft\uff08 –  {frac {a} {q}} cdot nright\uff09} \u96e2\u6563\u30d5\u30fc\u30ea\u30a8\u304c\u5909\u63db\u3055\u308c\u307e\u3057\u305f \u304b\u3089 f \uff08 \uff08 n \u3001 Q \uff09\uff09 \uff09\uff09 {displaystyle f\uff08\uff08n\u3001q\uff09\uff09} \u3002\u3053\u306e\u30d5\u30fc\u30ea\u30a8\u306b\u5909\u63db\u3055\u308c\u305f\u3053\u306e\u30d5\u30fc\u30ea\u30a8\u306b\u4ee5\u4e0b\u304c\u9069\u7528\u3055\u308c\u307e\u3059 Ff\uff08 n \u3001 Q \uff09\uff09 = \uff08 f \u2217 c\u2219\uff08 n \uff09\uff09 \uff09\uff09 \uff08 Q \uff09\uff09 = \u2211d\u2223qf (qd)de cd\uff08 n \uff09\uff09 {displaystyle f_ {f}\uff08n\u3001q\uff09=\uff08f*c_ {bullet}\uff08n\uff09\uff09\uff08q\uff09= sum _ {dmid q} fleft\uff08{frac {q} {d}}\u53f3\uff09cdot c_ {d}\uff08n\uff09} \u3068 f \uff08 \uff08 n \u3001 Q \uff09\uff09 \uff09\uff09 = 1q\u2211a=1qFf\uff08 a \u3001 Q \uff09\uff09 \u305d\u3046\u3067\u3059 (aq\u22c5n){displaystyle f\uff08\uff08n\u3001q\uff09\uff09= {frac {1} {q}} sum _ {a = 1}^{q} f_ {f}\uff08a\u3001q\uff09eleft\uff08{a} {q}} cdot night\uff09}} \u306e\u305f\u3081\u306b \u9006\u5909\u63db \u3002 [5] \u3053\u308c\u3089\u306e\u5909\u63db\u3067\u306f\u3001\u6700\u5927\u306e\u5171\u901a\u5206\u5272\u4f53\u306e\u5f62\u6210\u306b\u3088\u308b\u6c7a\u5b9a\u7684\u65b9\u7a0b\u5f0f\u306f\u3001\u6700\u7d42\u7684\u306b\u6b63\u306e\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u3092\u6301\u3064\u591a\u304f\u306e\u4fc2\u6570\u3092\u8003\u616e\u306b\u5165\u308c\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002 \u4f8b [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u6700\u5927\u516c\u7d04\u6570\uff1a (n,q)=\u2211a=1qe(aq\u22c5n)\u22c5\u2211d\u2223qcd(a)d{displaystyle\uff08n\u3001q\uff09= sum _ {a = 1}^{q} eleft\uff08{frac {a} {q}} cdot nright\uff09cdot sum _ {dmid q} {frac {c_ {d}\uff08a\uff09} {d}}}}} \u3053\u306e\u30d7\u30ec\u30bc\u30f3\u30c6\u30fc\u30b7\u30e7\u30f3\u306b\u3088\u308a\u3001\u305d\u3082\u305d\u3082\u6700\u5927\u306e\u5171\u901a\u4ed5\u5206\u3051\u306e\u5206\u6790\u7684\u7d99\u7d9a\u304c\u53ef\u80fd\u306b\u306a\u308a\u307e\u3059 n {displaystyle n} \u306e\u4e0a n \u2208 C{displaystyle nin mathbb {c}} \u5168\u4f53\u3068\u3057\u3066\u3002 [5] \u03c6(q)=\u2211a=1q(a,q)\u22c5e(\u2212aq){displaystyle varphi\uff08q\uff09= sum _ {a = 1}^{q}\uff08a\u3001q\uff09cdot eleft\uff08 –  {frac {a} {q}}\u53f3\uff09} \u3053\u308c\u306b\u7d9a\u3044\u3066\u3001\u5b9f\u969b\u306e\u90e8\u5206\u3068\u60f3\u50cf\u4e0a\u306e\u90e8\u5206\u306b\u5206\u5272\u3059\u308b\u4e09\u89d2\u95a2\u4fc2\u304c\u7d9a\u304d\u307e\u3059 \u2211a=1q(a,q)\u22c5cos\u2061(2\u03c0\u22c5aq)=\u03c6(q){displaystyle sum _ {a = 1}^{q}\uff08a\u3001q\uff09cdot cos left\uff082pi cdot {frac {a} {q}}\u53f3\uff09= varphi\uff08q\uff09quad} \u3068 \u2211a=1q(a,q)\u22c5sin\u2061(2\u03c0\u22c5aq)=0.{displaystyle quad sum _ {a = 1}^{q}\uff08a\u3001q\uff09cdot sin left\uff082pi cdot {frac {a} {q}}\u53f3\uff09= 0\u3002} \u5206\u5272\u95a2\u6570 \u03c3k{displaystyle sigma _ {k}} \u306e\u305f\u3081\u306e\u3053\u3068\u304c\u3067\u304d\u307e\u3059 0″>\u30e9\u30de\u30cc\u30b8\u30e3\u30f3\u306e\u5408\u8a08\u3092\u9023\u7d9a\u3068\u3057\u3066\u660e\u793a\u7684\u306b\u4f7f\u7528\u3057\u307e\u3059\u3002 [6] \u03c3k(n)=\u03b6(k+1)nk\u2211m=1\u221ecm(n)mk+1{displaystyle sigma _ {k}\uff08n\uff09= zeta\uff08k+1\uff09n^{k} sum _ {m = 1}^{infty} {frac {c_ {m}\uff08n\uff09} {m^{k+1}}}}}} \u306e\u6700\u521d\u306e\u5024\u306e\u8a08\u7b97 cm\uff08 n \uff09\uff09 {displaystyle c_ {m}\uff08n\uff09} \u300c\u5e73\u5747\u30b5\u30a4\u30ba\u300d\uff08\u5e73\u5747\u30b5\u30a4\u30ba\uff09\u306e\u5468\u308a\u306e\u5909\u52d5\u3092\u793a\u3057\u3066\u3044\u307e\u3059 z \uff08 k + \u521d\u3081 \uff09\uff09 nk{displaystyle zeta\uff08k+1\uff09n^{k}} \uff1a \u03c3k(n)=\u03b6(k+1)nk[1+(\u22121)n2k+1+2cos\u20612\u03c0n33k+1+2cos\u2061\u03c0n24k+1+\u22ef]{displaystyle sigma _ {k}\uff08n\uff09= zeta\uff08k+1\uff09n^{k}\u5de6[1+ {frac {\uff08-1\uff09^{n}} {2^{k+1}}}}+{frac {2cos {frac {2pi n} {3^{frac {frac {frac {3^}}}}}}}} {2cos {frac {pi n} {2}}} {4^{k+1}}+cdots\u53f3]} \u30e9\u30de\u30cc\u30b8\u30e3\u30f3\u30b9\u30de\u30e0\u306e\u4e00\u7a2e\u306e\u76f4\u4ea4\u6027\uff1abe  \uff08 n \uff09\uff09 {displaystyle eta\uff08n\uff09} \u6570\u306e\u7406\u8ad6\u7684\u306a1\u3064\u306e\u95a2\u6570\u3001\u3059\u306a\u308f\u3061\u3001\u6298\u308a\u305f\u305f\u307f\u64cd\u4f5c\u306e\u4e2d\u7acb\u8981\u7d20 \u03b7(n)={1(n=1)0(n\u2208N\u2216{1}).{displaystyle eta\uff08n\uff09= {begin {cases} 1quad\uff08n = 1\uff09\\ 0quad left\uff08nin mathbb {n} setminus lbrace 1rbrace right\uff09.end {cases}}}}}} \u6b21\u306b\u3001\u9006\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3092\u4ecb\u3057\u3066\u7d9a\u304d\u307e\u3059 n \u2208 Z\u2216 { 0 } \u3001 Q \u2208 N\uff1a {displaystyle nin mathbb {z} setminus lbrace 0rbrace\u3001qin mathbb {n} colon}  \u03b7((n,q))=1q\u2211a=1qcq(a)\u22c5e(aq\u22c5n){displaystyle eta\uff08\uff08n\u3001q\uff09\uff09= {frac {1} {q}} sum _ {a = 1}^{q} c_ {q}\uff08a\uff09cdot eleft\uff08{a} {q}} cdot nright\uff09}} \u3064\u307e\u308a\u3001\u6b63\u78ba\u306b\u306f\u6cd5\u5b9a\u984d\u304c\u6d88\u3048\u306a\u3044\u5834\u5408\u3001\u6570\u5b57\u306f n {displaystyle n} \u3068 Q {displaystyle q} \u30a8\u30a4\u30ea\u30a2\u30f3\u3002\u65b9\u7a0b\u5f0f\u306e\u53f3\u5074\u306b\u306f\u50241\u304c\u3042\u308a\u307e\u3059\u3002 \u30e8\u30eb\u30b0\u5144\u5f1f\uff1a \u5206\u6790\u756a\u53f7\u7406\u8ad6\u306e\u7d39\u4ecb \u3002 Springer\u3001Berlin\u3001Heidelberg\u3001New York 1995\u3001ISBN 3-540-58821-3\u3002  \u30b4\u30c3\u30c9\u30d5\u30ea\u30fc\u30cf\u30ed\u30eb\u30c9\u30cf\u30fc\u30c7\u30a3\uff1a \u30e9\u30de\u30cc\u30b8\u30e3\u30f3\uff1a\u5f7c\u306e\u4eba\u751f\u3068\u4ed5\u4e8b\u306b\u3088\u3063\u3066\u63d0\u6848\u3055\u308c\u305f\u4e3b\u984c\u306b\u95a2\u3059\u308b12\u306e\u8b1b\u7fa9 \u3002\u30a2\u30e1\u30ea\u30ab\u6570\u5b66\u5354\u4f1a\/\u30c1\u30a7\u30eb\u30b7\u30fc\u3001\u30d7\u30ed\u30d3\u30c7\u30f3\u30b91999\u3001ISBN 978-0-8218-2023-0\u3002  \u30b4\u30c3\u30c9\u30d5\u30ea\u30fc\u30cf\u30ed\u30eb\u30c9\u30cf\u30fc\u30c7\u30a3\u3001\u30a8\u30c9\u30ef\u30fc\u30c9\u30e1\u30a4\u30c8\u30e9\u30f3\u30c9\u30e9\u30a4\u30c8\uff1a \u6570\u5b57\u306e\u7406\u8ad6\u306e\u7d39\u4ecb \u3002\u7b2c5\u7248\u3002\u30aa\u30c3\u30af\u30b9\u30d5\u30a9\u30fc\u30c9\u5927\u5b66\u51fa\u7248\u5c40\u3001\u30aa\u30c3\u30af\u30b9\u30d5\u30a9\u30fc\u30c91980\u3001ISBN 978-0-19-853171-5\u3002  \u30b8\u30e7\u30f3\u30fb\u30ce\u30c3\u30d7\u30de\u30c3\u30cf\u30fc\uff1a \u62bd\u8c61\u5206\u6790\u756a\u53f7\u7406\u8ad6 \u3002\u65b0\u7248\u3002 Dover Publications\u30012000\u3001ISBN 0-486-66344-2\u3002  Srinivasa Ramanujan\uff1a \u7279\u5b9a\u306e\u4e09\u89d2\u6cd5\u3068\u6570\u5b57\u306e\u7406\u8ad6\u306b\u304a\u3051\u308b\u305d\u308c\u3089\u306e\u30a2\u30d7\u30ea\u30b1\u30fc\u30b7\u30e7\u30f3\u306b\u3064\u3044\u3066 \u3002\u306e\uff1a \u30b1\u30f3\u30d6\u30ea\u30c3\u30b8\u54f2\u5b66\u5354\u4f1a\u306e\u53d6\u5f15 \u3002 \u30d0\u30f3\u30c9  22 \u3001 \u3044\u3044\u3048\u3002  15 \u30011918\u5e74\u3001 S.  259\u2013276 \u3002  Srinivasa Ramanujan\uff1a \u7279\u5b9a\u306e\u7b97\u8853\u95a2\u6570\u306b\u3064\u3044\u3066 \u3002\u306e\uff1a \u30b1\u30f3\u30d6\u30ea\u30c3\u30b8\u54f2\u5b66\u5354\u4f1a\u306e\u53d6\u5f15 \u3002 \u30d0\u30f3\u30c9  22 \u3001 \u3044\u3044\u3048\u3002  9 \u30011916\u5e74\u3001 S.  159\u2013184 \u3002  Srinivasa Ramanujan\uff1a \u96c6\u3081\u3089\u308c\u305f\u8ad6\u6587 \u3002\u30a2\u30e1\u30ea\u30ab\u6570\u5b66\u5354\u4f1a\/\u30c1\u30a7\u30eb\u30b7\u30fc\u3001\u30d7\u30ed\u30d3\u30c7\u30f3\u30b92000\u3001ISBN 978-0-8218-2076-6\u3002  \u30ed\u30d0\u30fc\u30c8\u30c1\u30e3\u30fc\u30eb\u30ba\u30f4\u30a9\u30fc\u30f3\uff1a Hardy-Littlewood\u30e1\u30bd\u30c3\u30c9 \u3002\u7b2c2\u7248\u200b\u200b\u3002\u30b1\u30f3\u30d6\u30ea\u30c3\u30b8\u5927\u5b66\u51fa\u7248\u5c40\u3001\u30b1\u30f3\u30d6\u30ea\u30c3\u30b81997\u3001ISBN 0-521-57347-5\u3002  Wolfgang Schramm\uff1a \u6700\u5927\u306e\u5171\u901a\u9664\u6570\u306e\u6a5f\u80fd\u306e\u30d5\u30fc\u30ea\u30a8\u5909\u63db \u3002\u306e\uff1a Integers\uff1aElectronic Journal of Combinatorical Number Theory \u3002 \u30d0\u30f3\u30c9  8 \u3001 \u3044\u3044\u3048\u3002  50 \u30012008\u5e74\uff08 emis.de [PDF]\uff09\u3002  Ivan Matvevitch Vinogradov\uff1a \u6570\u5b57\u306e\u7406\u8ad6\u306b\u304a\u3051\u308b\u4e09\u89d2\u6e2c\u5b9a\u306e\u65b9\u6cd5 \u3002\u30ed\u30b7\u30a2\u8a9e\u304b\u3089\u7ffb\u8a33\u3055\u308c\u3001\u30af\u30e9\u30a6\u30b9\u30fb\u30d5\u30ea\u30fc\u30c9\u30ea\u30c3\u30d2\u30fb\u30ed\u30b9\u3068\u30a2\u30f3\u30fb\u30a2\u30b7\u30e5\u30ea\u30fc\u30fb\u30c0\u30d9\u30f3\u30dd\u30fc\u30c8\u306b\u3088\u3063\u3066\u6ce8\u91c8\u304c\u4ed8\u3051\u3089\u308c\u307e\u3057\u305f\u3002\u30cb\u30e5\u30fc\u30e8\u30fc\u30af\u3001\u30c9\u30fc\u30d0\u30fc2004\u3002  \u2191  \u30e9\u30de\u30cc\u30b8\u30e3\u30f3\uff081916\uff09\u3002  \u2191  \u30f4\u30a9\u30fc\u30f3\uff081997\uff09\u3002  \u2191  \u5144\u5f1f\uff081995\uff09p\u300220\u3002  \u2191  \u5144\u5f1f\uff081995\uff09\u88dc\u984c1.3.1\u3002  \u2191 a b c  Schramm\uff082008\uff09\u3002  \u2191  E.\u30af\u30ec\u30c3\u30c4\u30a7\u30eb\uff1a \u756a\u53f7\u7406\u8ad6 \u3002 Veb Deutscher Verlag Der Sciences\u3001Berlin 1981\u3001 S.  130 \u3002         (adsbygoogle = window.adsbygoogle || 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