TitleHow should we bet on prime number dice? (Analytic NumberTheory : Arithmetic Properties of Transcendental Functionsand their Applications)
Author(s) å°å·, è£ä¹
Citation æ°ç解æç 究æè¬ç©¶é² (2014), 1898: 16-27
Issue Date 2014-05
URL http://hdl.handle.net/2433/195899
Right
Type Departmental Bulletin Paper
Textversion publisher
Kyoto University
How should we bet on prime number dice?
倧éªå€§åŠå€§åŠé¢çåŠç ç©¶ç§ å°å·è£ä¹
Hiroyuki OgawaDepartment of Mathematics, Osaka University
\S 1. åºè«
Dirichlet ã®çŽ æ°å®çããïŒä»»æã®æŽæ° $d$ ã«å¯ŸããŠïŒæ³ $d$ ã«é¢ããã©ã®æ¢çŽå°äœé¡ã«ãåãå²å
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\S 2. åæ©
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ãšãã« 6ã®åæ°ã«ãªã確ç㯠1/4ãšèãããïŒ$\sum_{6|d}g_{d}(x)\sim 1/4\pi(x)$ ãšèããããŸãïŒåé¡ãåçŽåã㊠$\#\{p\leq x|6|p+p_{n}\}/\pi(x)$ ãèããŠã¿ãŸãïŒ1/2ã«ãªããšæãããŸããïŒ$x=10^{7}$ ã§èšç®ãããš 55.9% 㧠50% ã«ãªãããã«ãªãïŒé¢çœã話ã«ãªããããããªããïŒå€ããã調ã¹ãããŠããå¯èœæ§ãããã®ã§ïŒæç®ãååã«èª¿ã¹ãæ¹ãããã§ãããïŒ
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\S 3. ååæ¡ä»¶ä»ãçŽ æ°ååž
èªç¶æ° $n$ ã«å¯ŸããŠïŒ$\mu(n)$ 㧠$n$ ãçŽ æ°ã§ãã確çãšããïŒ$\mu(n)$ ã¯ïŒæçå°é¢æ°ã«å¯ŸããŠçŽ æ°äžã§é¢æ°ã®å€ã®åããšãç©åæ žïŒçŽ æ°ã«å°ãã〠Dirac è¶ é¢æ°ã§å®çŸ©ãïŒæçå°é¢æ°ç©ºéäžã®æ±é¢æ°ã®ç©ºéã§è¿äŒŒãèããããšã«ãªãïŒç ©éã«ãªãã®ã§çŽæçãªèª¬æã«ãšã©ããïŒã€ãŸãããã§èšã確ç
ã¯ïŒæçåºéã«ãããäºè±¡ã®èµ·ãã床æ°ãšåºéé·ã®æ¯ãšããïŒ$\mu(n)$ ã«ã€ããŠã¯ $\pi(x)=\int_{1}^{x}\mu(n)dn$
ã§ïŒçŽ æ°å®çãã $\mu(n)\sim\log(n)^{-1}$ ãšãªãïŒèªç¶æ° $n$ ã«ã€ããŠïŒ$n$ ãã倧ããæå°ã®çŽ æ°ã np$(n)$
(次ã®çŽ æ°ïŒnext prime) ã§è¡šãïŒæ¬¡ã®çŽ æ°ãšã®å·®ã gap$(n)(= np(n)-n)$ ã§è¡šãïŒ$d$ ãèªç¶æ°ãš
ããïŒèªç¶æ° $n$ ã«ã€ããŠïŒgap $(n)=d$ ãšãªã確çã $\mu(n|$ gap$(n)=d)$ ã§è¡šãïŒçŽ æ° $n$ ã«ã€ããŠïŒga$P$ $(n)=d$ ãšãªã確çã $\mu_{p}(n|$ gap$(n)=d)$ ã§è¡šãïŒèªç¶æ° $n$ ã«ã€ããŠïŒ$n$ ãçŽ æ°ã§ gap$(n)=d$ ãš
ãªã確çã $\mu(n, d)$ ã§è¡šãïŒèªç¶æ° $n$ ãçŽ æ°ã§ããäºè±¡ãšïŒgap$(n)$ ã®å€ã«é¢ããäºè±¡ãç¬ç«ãªãïŒ$\mu(n|$ gap$(n)=d)=\mu_{p}(n|$ gap$(n)=d),$ $\mu(n, d)=\mu(n)\mu_{p}(n|$ gap$(n)=d)$ ãæãç«ã€ïŒã©ã®èªç¶æ°ã«å¯ŸããŠãçŽ æ°ãšãªãäºè±¡ãç¬ç«ãªãïŒ$\mu(n|$ gap$(n)=d)= \mu(n+d)\prod_{t=1}^{d-1}(1-\mu(n+t))(n+1,$
$\cdots,$
$n+d-1$ ãåææ°ã§ $n+d$ ãçŽ æ°ãšãªã確ç) ãæãç«ã€ïŒåå倧ãã $n$ ã«å¯ŸããŠïŒ$\mu(n)\sim\log(n)^{-1}$ã§è¿äŒŒãããšïŒ$\mu(n|gap(n)=d)\sim\log(n)^{-1}(1-\log(n)^{-1})^{d-1}$ , åŸã£ãŠ $\mu(n, d)\sim\log(n)^{-2}$ ãšãª
ãïŒçŽ æ°ã®ééã«é¢ãã Hardy-Littlewood ã®äºæ³ $\pi_{d}(x)\sim c_{d}x\log(x)^{-2}$ (ããã§ïŒ$\pi_{d}(x)$ 㧠$x$
以äžã®çŽ æ° $n$ 㧠$n+d$ ãçŽ æ°ãšãªããã®ã®åæ°ãè¡šãïŒ$c= \prod_{p\geq 3}p(p-2)/(p-1)^{2}=0.66016\cdots,$
$c_{d}=2c \prod_{3\leq p|d}(p-1)/(p-2)$ ãšãã) ã«ãããš (次ç¯åç §), 倧éæã ãå·®åãåãããšã§ïŒ$n$ ãš
$n+d$ ãå ±ã«çŽ æ°ãšãªã確çã®äž»èŠé 㯠$c_{d}\log(n)^{-2}$ ãšãªãïŒHardy-Littlewood ã®äºæ³ã§ã¯ $n$ ãš
17
$n+d$ ã®éã®çŽ æ°ã®æç¡ãèæ ®ããŠããªãã®ã§ïŒ$n$ ãš $n+d$ ã®éã«çŽ æ°ã®ãªãç¶æ³ $\mu(n, d)$ ã«ã€ããŠ
çŽæ¥è¿°ã¹ãŠã¯ããªããïŒäºæ³ã«éããæšè«ã $n$ ãš $n+d$ ãå ±ã«çŽ æ°ã§ãã®éã«ä»ã®çŽ æ°ãããå Žåã«çšããã°ãã®ç¢ºç㯠$O(\log(n)^{-3})$ ã§è©äŸ¡ãããã®ã§ïŒ$\mu(n, d)\sim c_{d}\log(n)^{-2}$ ãšãªãïŒå ã«èŠã$\mu(n, d)\sim\log(n)^{-2}$ ãšã¯äž»èŠé ãå®æ°å $(c_{d}$ å $)$ éã£ãŠããïŒèªç¶æ°ãçŽ æ°ã§ããäºè±¡ããã¹ãŠç¬ç«ã§ãããšä»®å®ããããšã«èµ·å ããïŒåœããåã®ããšã ãïŒé£ç¶ 2èªç¶æ°ã®çµããšãã«çŽ æ°ãšãªãäºè±¡ $(d=1$ ã®äŸ $)$ 㯠{2, 3} 以å€ã«ãªãã®ã§ç¢ºç㯠$0$ ã§ïŒ$\mu(n)\mu(n+1)\sim\log(x)^{-2}$ ã¯ããè¿äŒŒãšã¯èšããªãïŒå°ãªããšãååé¢ä¿ã«ãã食ããããã¹ãã§ããïŒHardy-Littlewood ã®äºæ³ã¯ïŒååé¢ä¿ã«ããçã®ããã«ïŒæ¯èŒçå°ããåºéã«ãããæ¢çŽå°äœã®ååžã®åããèæ ®ããŠããïŒå®éšå€ããšãŠãè¯ãè¿äŒŒããïŒãšãããïŒ$n$ ãçŽ æ°ã§ããããšãä»®å®ããªã $\mu(n|$ gap$(n)=d)$ ã«ã€ããŠå®éš
ãããšïŒ$\log(n)^{-1}(1-\log(n)^{-1})^{d-1}$ ãæå€ã«è¯ãè¿äŒŒãšãªã£ãŠããïŒäœãïŒå®éšèŠ³å¯ãç¶ãããšã©ã¡ãã®è©äŸ¡ãå°ãéãªæ°ãããïŒå®éšå€ãšã®æ¯ã¯ïŒã©ã¡ããåããããªæåãããŠããŠïŒ$d$ ã«é¢ããŠããçš®ã®æžè¡°ãäžããé ã欲ãããªãïŒæ³ $m$ ã«é¢ããæ¢çŽå°äœé¡ $\alpha$ ãšæ³ $mâ$ ã«é¢ããæ¢çŽå°äœé¡ $\beta$ ãåãïŒèªç¶æ° $n$ ã«ã€ããŠïŒnp$(n)\in\beta$
ãšãªã確çã $\mu(n| np(n)\in\beta)$ ãšããïŒå°äœé¡ $\alpha$ ã«å±ããçŽ æ° $n$ ã«ã€ããŠïŒ$np(n)\in\beta$ ãšãªãæ¡ä»¶ä»ã確çã $\mu_{p}(n;\alphaarrow\beta)$ ãšããïŒèªç¶æ° $n$ ã«å¯ŸããŠïŒ$n$ ãã倧ãã $\beta$ ã®ãã¹ãŠã®å ã䞊ã¹$n<n_{1}<n_{2}<n_{3}<\cdots(n_{i}\in\beta, n_{i+1}=n_{i}+mâ, n_{1}-mâ\leq n),$ $d_{i}=n_{i}-n$ ãšããïŒ$\mu(n|np(n)\in\beta)$ ã¯$\mu(n|$ gap$(n)=d_{i})$ ã®å $\sum_{i\geq 1}\mu(n|$ gap$(n)=d_{i})$ ã«çããïŒå ã«ïŒçŽ æ°ã§ãããã©ããïŒäºè±¡ã®ç¬ç«æ§ãä»®å®ã§ããªãããšã«è§ŠãããïŒããã§ã®äºè±¡ã¯æåã§ããããåçŽã«åãåã£ãŠããïŒäœãïŒæ¥µããŠåæã®é ãçŽæ°ã§ããïŒæ¡ä»¶ä»ã確ç $\mu_{p}(n;\alphaarrow\beta)$ ã«ã€ããŠãåæ§ã«çŽæ° $\sum_{i\geq 1}\mu_{p}(n|$ gap$(n)=d_{i})$
ã§è¡šãããïŒ$n$ ãçŽ æ°ã§ããããšãã $\mu(n|gap(n)=d_{i})$ ã®ä»£ããã« $\mu_{p}(n|gap(n)=d_{i})$ ã®åãšãªãïŒ$n\in\alpha$ ã®æ¡ä»¶ãè¡šç«ã£ãŠçŸããŠãªããïŒ$\beta$ ã®å ã®å $n_{1}<n_{2}<\cdots$ ãèããéã«ãã®æ¡ä»¶ã圱é¿ãäž
ããïŒ$n_{i}$ ãæ³ $m$ ãšçŽ ã§ãªããšã $n_{i}$ ã¯çŽ æ°ã§ãªãããšã«ãªãã®ã§ïŒ$\mu_{p}(n|gap(n)=d_{i})(d_{i}=n_{i}-n)$ã $O$ ãšæã£ãŠãããïŒãããŸã§ã確ççãªæå³ã§ãã®é ãæ®ããŠããããšãã§ãããïŒååé¢ä¿ã«ãã食ãããã Hardy-Littlewoo$d$ ã®äºæ³ãèžãŸããè°è«ããã®åé¡ã«å¯ŸããŠå±éããã®ã ããïŒ
$n_{i}$ ã $m$ ãšçŽ ã§ãªããšããã¯é€ãã¹ãã§ãããïŒæ¡ä»¶ä»ã確ç $\mu_{p}(n;\alphaarrow\beta)$ ã®æ¯ãèãã«ã€ããŠïŒãããŸã§ã®èšå®ã§èšŒæã§ããããšãããïŒ
å®ç 3.1 $m$ ãš $mâ$ ã®äžæ¹ã 2ã®å¹ã§ä»æ¹ã 4ã§å²ãåããªããšãïŒ$\mu_{p}(n;\alphaarrow\beta)\sim\frac{1}{\varphi(mâ)}$
å°ãéã«å®çŸ©ãã確çã«å¯ŸããŠäž»èŠé ã®æ¯ãèããè¿°ã¹ããã®ã§ïŒäœã ããšãŠãææ§ïŒå°ãè¡šçŸãå€ããïŒæ³ $m$ ã«é¢ããæ¢çŽå°äœé¡ $\alpha$ , æ³ $mâ$ ã«é¢ããæ¢çŽå°äœé¡ $\beta$ ãšïŒæ£ã®å®æ° $x$ ã«å¯Ÿã
ãŠïŒ$\pi_{\alpha,\beta}(x)=\#\{p<x|p$ ã¯çŽ æ°ïŒ$p\in\alpha,$ $np(p)\in\beta\},$ $\pi_{\alpha}(x)=\#\{p<x|p$ ã¯çŽ æ°ïŒ$P\in\alpha\}$ ãšããïŒæ¡ä»¶ä»ã確ç $\mu_{p}(n;\alphaarrow\beta)$ ã«é¢ä¿ããã®ãïŒååæ¡ä»¶ä»ãçŽ æ°ååž $\pi_{\alpha,\beta}(x)/\pi_{\alpha}(x)$ ã§ããïŒ
å®ç 3.2 $m$ ãš $mâ$ ã®äžæ¹ã 2ã®å¹ã§ä»æ¹ã 4ã§å²ãåããªããšãïŒ$\pi_{\alpha,\beta}(x)\sim\frac{1}{\varphi(mmâ)}\pi(x)$
å®çããã³ãã®èšŒæã«ãããŠïŒèª€å·®é ã®è©³çŽ°ãè¿°ã¹ãŠããªãïŒåŸã§è¿°ã¹ãããã«ïŒæ³ã«é¢ããæ¡ä»¶ãªãã«ïŒHardy-Littlewood ã®äºæ³ã®ããšã§ $\pi_{\alpha,\beta}(x)/\pi(x)arrow 1/\varphi(m)\varphi(mâ)(xarrow\infty)$ ãåŸãããïŒ
äžã®å®çãšã®éãã¯ïŒèª€å·®é ã®è©äŸ¡ã«ããïŒãã®æå³ã§ã¯èª€å·®é ã䞹念ã«è©äŸ¡ããå¿ èŠããããïŒå®çã«ãããæ³ã«é¢ããæ¡ä»¶ã®ããšã§ $\pi(x)$ ã $\pi_{\alpha}(x)$ ã®çŽ æ°å®çãªã©ã§ç¥ãããŠãã誀差è©äŸ¡ããã®ãŸãŸäœ¿ãããšãã§ããïŒããã§ã¯ãã¹ãŠçç¥ããïŒ
$\alphaâ=\{n\in\alpha|(n, m_{0})=1\},$ $\betaâ=\{n\in\beta|(n, mo)=1\}$ ãïŒæ³ $m_{0}(=1cm(m, mâ))$ ã®æ¢çŽå°äœé¡ã®é亀åå $\alphaâ=\alpha_{1}\cup\cdots\cup\alpha_{s},$ $\betaâ=\beta_{1}\cup\cdots\cup\beta_{t}$ ã«å解ããïŒ$\mu_{p}(n;\alphaarrow\beta)=\sum_{i,j}\mu_{p}(n;\alpha_{i}arrow\beta_{j})$ ,$\pi_{\alpha,\beta}(x)=\sum_{i,j}\pi_{\alpha_{i},\beta_{j}}(x)+O(1)$ ( $m_{0}$ ã®çŽ å åãæ°ãããæ°ããªãã£ãã) ãšè¡šãã å°ã倧é
æã ãïŒ$\pi_{\alpha_{i},\beta_{j}}(x)$ ã¯æž¬åºŠ $\mu_{p}(n;\alpha_{i}arrow\beta_{j})dn$ ã«é¢ããåºé $(0, x]$ ã®äœç©ã§ããïŒ$s\cross t$ åã®æž¬åºŠ
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$\mu_{p}(n;\alpha_{i}arrow\beta_{j})dn$ ã¯ãã¹ãŠå°ãç°ãªãã®ã§ïŒ$\sum_{i,j}\pi_{\alpha_{i},\beta_{j}}(x)$ ã®åã¯ïŒæž¬åºŠã®å $\sum_{i,j}\mu_{p}(n;\alpha_{i}arrow\beta_{j})dn$
ã«é¢ããåºé $(0, x]$ ã®äœç©ã«çããïŒ$mâ$ ã 2å¹ïŒ$m$ ã 4ã§å²ãåããªãèªç¶æ°ãšããïŒ$T:n\mapsto n+m_{1}(m_{1}=lcm(m, 2))$ ãšããïŒå¹³
è¡ç§»å $T$ ã¯ïŒ$(\mathbb{Z}/m\mathbb{Z})^{*},$ $(\mathbb{Z}/mâ\mathbb{Z})^{*},$ $(\mathbb{Z}/m_{0}\mathbb{Z})^{*}$ ã®äžã«çœ®æãåŒãèµ·ããïŒããã眮æã $T$ ã§è¡šãïŒ$\mu_{p}(n;\alpha_{i}arrow\beta_{j})$ 㯠$n\in\alpha_{i}$ ãš $\beta_{j}$ ã®å ã®å·®ã§å®ãŸãã®ã§ïŒ$\mu_{p}(T(n);T(\alpha_{i})arrow T(\beta_{j}))\sim\mu_{p}(n;\alpha_{i}arrow\beta_{j})$ ãš
ãªãïŒçœ®æ $T$ ã® $(\mathbb{Z}/m\mathbb{Z})^{*}$ ãžã®äœçšã¯æç眮æãªã®ã§ïŒ$T$ 㯠$\{\alpha_{1}, \cdots, \alpha_{s}\}$ ã®äžã«çœ®æãåŒãèµ·ããïŒ$\mu_{p}(n;\alphaarrow T(\beta))\sim\mu_{p}(T(n);T(\alpha)arrow T(\beta))=\sum_{i,j}\mu_{p}(T(n);T(\alpha_{i})arrow T(\beta_{j}))\sim\sum_{i,j}\mu_{p}(n;\alpha_{i}arrow\beta_{j})$
$=\mu_{p}(n;\alphaarrow\beta)$ ãæãç«ã€ïŒçœ®æ $T$ ã® $(\mathbb{Z}/mâ\mathbb{Z})^{*}$ ãžã®äœçšã¯ãã¹ãŠã®æ¢çŽå°äœé¡ (å¥æ°) ãæž¡ãäœæ°$\varphi(mâ)(=mâ/2)$ ã®å·¡å眮æãªã®ã§ïŒ$\mu_{p}(n;\alphaarrow\beta)$ 㯠$\beta\in(\mathbb{Z}/mâ\mathbb{Z})^{*}$ ã«äŸããªãïŒåŸã£ãŠ $\pi_{\alpha,\beta}(x)$ ã®
äž»èŠé 㯠$\beta\in(\mathbb{Z}/mâ\mathbb{Z})^{*}$ ã«äŸããªãïŒ$\sum_{\beta\in(\mathbb{Z}/mâ\mathbb{Z})^{r}}\pi_{\alpha,\beta}(x)\sim\pi_{\alpha}(x)$ ãªã®ã§ $\pi_{\alpha,\beta}(x)\sim\pi_{\alpha}(x)/\varphi(mâ)$
ãšãªãïŒDirichlet ã®çŽ æ°å®çãã $\pi_{\alpha,\beta}(x)\sim\pi(x)/\varphi(mmâ)$ ãåŸãïŒ$m$ ã 2å¹ïŒ$mâ$ ã 4ã§å²ãåããªãèªç¶æ°ãšããïŒ$\pi_{\alpha,\beta}(x)(\alpha\in(\mathbb{Z}/m\mathbb{Z})^{*}, \beta\in(\mathbb{Z}/mâ\mathbb{Z})^{*})$ ã®äž»
èŠé 㯠$\alpha$ ã«äŸããªãïŒ$\sum_{\alpha}\pi_{\alpha,\beta}(x)=\pi_{\beta}(x)+O(1)(m_{0}$ ã®çŽ å åãšïŒ$x$ ãã倧ãã $x+mâ$ ãè¶ããªãç¯å²ã®çŽ æ°ãæ°ãããæ°ããªãã£ãã) ãªã®ã§ïŒ$\pi_{\alpha,\beta}(x)\sim\pi_{\beta}(x)/\varphi(m)\sim\pi(x)/\varphi(mmâ)$ ãåŸãïŒ
\S 4. çŽ æ°ééã«é¢ãã Hardy-Littlewood ã®äºæ³
æ£ã®å¶æ° $d$ ã«å¯ŸããŠïŒ$\pi_{d}(x)=\#\{n<x|n,$ $n+d$ ãå ±ã«çŽ æ° $\}$ ãšããïŒHardy-Littlewood 㯠$\pi_{d}(x)$
ã®è©äŸ¡ã«ã€ããŠæ¬¡ã®ããã«äºæ³ããïŒ
äºæ³ 4.1 (Hardy-Littlewood) $\pi_{d}(x)\sim c_{d}\frac{x}{\log(x)^{2}}$ äœãïŒ$c= \prod_{p\geq 3}\frac{p(p-2)}{(p-1)^{2}},$ $c_{d}=2c \prod_{3\leq p|d}\frac{(p-1)}{(p-2)}$
G.H.Hardy-E.M.Write â An Introduction to the Theory of Numbersâ ã«åŸã£ãŠãã®äºæ³ã«è³ã
æšè«ã玹ä»ããïŒ$N_{x}= \prod_{p\leq\sqrt{x}}p$ ãšããïŒæ£ã®å®æ° $X$ ã«å¯Ÿã㊠$X$ 以äžã®èªç¶æ°ã§ $N_{x}$ ãšçŽ ãªãã®ã®åæ°ã $S(X)$ ãšããïŒãã®ãšã $S(N_{x})= \varphi(N_{x})=N_{x}\prod_{p\leq\sqrt{x}}(1-1/p)$ ãšãªãïŒMertens ã®å®çãã$\prod_{p\leq\sqrt{x}}(1-1/p)\sim e^{-\gamma}/\log\sqrt{x}=2e^{-\gamma}/\log(x)$ ãæãç«ã€ã®ã§ïŒ$S(N_{x})/N_{x}\sim 2e^{-\gamma}/\log(x)$ ãåŸãïŒ$\sqrt{x}$ ãã倧ãã $x$ ãè¶ããªãçŽ æ°ã¯ $N_{x}$ ãšçŽ ãªèªç¶æ°ãªã®ã§ïŒ$S(x)=\pi(x)-\pi(\sqrt{x})\sim x/\log(x)$ ãæãç«ã€ïŒ$S(x)/x\sim 1/\log(x)$ ãåŸãïŒ$N_{x}$ ãã倧ãã $X$ ã«å¯ŸããŠã¯ïŒ$X$ 以äžã®èªç¶æ°ã®äžã§ $N_{x}$ ãšçŽ ãªãã®ã®å²å㯠$S(X)/X\sim 2e^{-\gamma}/\log(x)$ ãªã®ã«å¯ŸããŠïŒ$x$ 以äžã®èªç¶æ°ã®äžã§ $N_{x}$ ãšçŽ ãªãã®ã®å²å㯠$S(x)/x\sim 1/\log(x)$ ã§ïŒéåžž $($ ? $)$ ããè¥å¹²å°ãªãã® $e^{\gamma}/2(=0.89\cdots)$ åã§ããïŒãªããšãæªããè©äŸ¡ã ãäœæ¥ä»®å®ãšããŠèªããŠãããŠïŒ$X$ 以äžã® $N_{x}$ ãšçŽ ãªèªç¶æ° $n$ 㧠$n+d$ ã $N_{x}$ ãšçŽ ãªãã®ã®åæ°$S_{d}(X)$ ã®è©äŸ¡ã«ç§»ãïŒ$p\leq\sqrt{x}$ ã«å¯Ÿã㊠$n\not\equiv O,$ $-dmod p$ ãªã®ã§ïŒ$S_{d}(N_{x})= \prod_{p/d}^{J}(p-2)\prod_{p|d}(p-1)$
ãšãªãïŒããã§ïŒ$\prodâ$ ã¯ã以äžã®å¥çŽ æ° $p$ ããããç©ãšããïŒ$N_{x}$ ãšçŽ ãªèªç¶æ°ã®å²åã«é¢ããã¡ãã£ãšæªããè°è«ããïŒ$N_{x}$ ãšçŽ 㪠2ã€ã®èªç¶æ°ã®çµã«èšåãã $S_{d}(x)$ ã«ã€ããŠïŒå®çŽã«å²åã®æ¯çã $(e^{\gamma}/2)^{2}$ ãšã㊠$S_{d}(x)/x\sim S_{d}(N_{x})/N_{x}\cross(e^{\gamma}/2)^{2}$ ãšèããããïŒ$e^{\gamma}/2 \sim 2\prodâ(1-1/p)^{-1}/\log(x)^{2}$
ãªã®ã§ïŒ$S_{d}(x)/x \sim 2\prodâ(1-2/p)/(1-1/p)^{2}\prod_{3\leq p|d}(p-1)/(p-2)/\log(x)^{2}\sim c_{d}/\log(x)^{2}$ ãåŸãïŒãšããã§ïŒ$S_{d}(x)=\pi_{d}(x)-\pi_{d}(\sqrt{x})$ ãªã®ã§ $\pi_{d}(x)=S_{d}(x)+S_{d}(\sqrt{x})+S_{d}(\sqrt[4]{x})+S_{d}(\sqrt[8]{x})+\cdots$ ãšãªãïŒ$S_{d}(\sqrt{x})\sim c_{d}\sqrt{x}/\log(\sqrt{x})^{2}=O(\sqrt{x}/\log(x)^{2})$ ãªã®ã§ïŒ$S_{d}(\sqrt{x}),$ $S_{d}(\sqrt[4]{x}),$ $S_{d}(\sqrt[8]{x}),$ $\cdots$ ã¯èª€å·®é ãšã¿ãªããïŒ$\pi_{d}(x)\sim S_{d}(x)\sim c_{d}x/\log(x)^{2}$ ãšãªãïŒ
Hardy-Littlewood ã®äºæ³ã«è³ã£ãäžã®æšè«ã§ã¯ïŒé¢ãã倧ãã $x$ ãè¶ããªãçŽ æ°ã®åæ°ãïŒåå倧ããç¯å²ã«ããã $N_{x}(= \prod_{p\leq\sqrt{x}}p)$ ãšçŽ ãªèªç¶æ°ã®å²åã® $e^{\gamma}/2(=0.89)$ åãšæšå®ãïŒçŽ æ°ã®2ã€çµãæ°ããå Žåã¯åçŽã« $(e^{\gamma}/2)^{2}$ åãšããïŒçŽ æ° 3 ã£çµã«å¯ŸããŠã¯ $(e^{\gamma}/2)^{3}$ åïŒ4ã€çµã«å¯ŸããŠã¯ $(e^{\gamma}/2)^{4}$ åãšæšå®ã§ããïŒãããŠïŒMertens ã®å®çã䜿ã£ãŠè©äŸ¡ããŠããéçšã§ $e^{\gamma}/2$ ã²ãšã€ã«ã€ã $\log(x)^{-1}$ ãã²ãšã€ãã€çŸããïŒ$n(\leq x)$ ãš $n+d$ ããšãã«çŽ æ°ã®ç¶æ³ã§ïŒ$n$ ã®æ¬¡ã®çŽ æ°ã $n+d$
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ã§ãªãå ŽåãèããïŒ$n$ ãš $n+d$ ã®éã«ä»ã®çŽ æ°ã幟ã€ãå ¥ãç¶æ³ãèããŠè©äŸ¡ããã®ã ãïŒéã«çŽ æ°ãã²ãšã€å ¥ãããšã«è©äŸ¡ã $\log(x)^{-1}$ åãããïŒãã®ãã㪠$n$ ã®åæ°ã¯ $O(x/\log(x)^{3})$ ã§è©äŸ¡
ãããïŒ$\tilde{\pi}_{d}(x)=\#\{n<x|n$ ã¯çŽ æ°ïŒga$P$ $(n)=d\}$ ãšãããšïŒ$\overline{\pi}_{d}(x)\sim\pi_{d}(x)\sim c_{d}x/\log(x)^{2}$ ãšãªãïŒ
åç¯ã§ïŒèªç¶æ° $n$ ã«ã€ããŠïŒ$n$ ãçŽ æ°ã§ gap$(n)=d$ ãšãªã確çã $\mu(n, d)$ ãšãããïŒ$\tilde{\pi}_{d}(x)$ ã¯æž¬
床 $\mu(n, d)$䌜ã«ãããåºé $(0, x]$ ã®äœç©ã«çããïŒHardy-Littlewood ã®äºæ³ã®ããšã§ïŒåå倧ãã $x$ ã«ãã㊠$\tilde{\pi}_{d}(x)$ ã®å·®åãåããš $\mu(n, d)\sim c_{d}/\log(n)^{2}$ ãåŸãïŒ
\S 5. æ°å€å®éšãšäºæž¬å€ã®æ¯èŒ
æ°å€å®éšãšäºæž¬å€ã®èšç®ïŒãããã®æ¯èŒã«ã€ããŠè¿°ã¹ãïŒ\S 3ã§èŠãããã« $m$ ãš $mâ$ ã®æå°å ¬åæ°ã
æ³ãšããæ¢çŽå°äœé¡éã®ååžã®åã«ã§ããã®ã§ïŒ$m=mâ$ ã§èããã°ããïŒ$\alpha,$ $\beta\in(\mathbb{Z}/m\mathbb{Z})^{*}$ ãšããïŒ
çŽ æ° $p\in\alpha$ ã®äžã§ np$(p)\in\beta$ ãšãªããã®ã®å²å $\varpi_{\alpha,\beta}(x;xâ)=(\pi_{\alpha,\beta}(xâ)-\pi_{\alpha,\beta}(x))/(\pi_{\alpha}(xâ)-\pi_{\alpha}(x))$
$(0\leq x<xâ)$ ãååæ¡ä»¶ä»ãçŽ æ°ååžãšåŒã¶ïŒ$2^{28}$ 以äžã®çŽ æ°ãçšæãïŒ$\varpi_{\alpha,\beta}(0;2^{e})(e=20, \cdots, 28)$
ãèšç®ããïŒnp$(2^{e})(e=20, \cdots, 44)$ ãã $10^{7}$ åãã€çŽ æ°ãçšæãïŒãããã $10^{6}$ åãã€ã® 10ã®åº
éã«åå²ãïŒããããã®åºé $(x<p\leq xâ)$ ããšã« $\varpi_{\alpha,\beta}(x;xâ)$ ãèšç®ããïŒ$2^{20}$ 㯠10é² 7æ¡ïŒ$2^{28}$ ã¯
10é² 9æ¡ïŒ$2^{44}$ 㯠10é² 14æ¡ã®æ°ã§ããïŒæ³ $m$ 㯠$10^{6}$ ã«æ¯ã¹ãŠååã«å°ããæ°ã§ãããªããããã§
ãããã®ã ãïŒååé¢ä¿ã§ç¯ããããã®ã ããå¶æ°ã«ãã¹ãã§ïŒããã§ã¯ $m=4,8,6,10,14,30$ ã
ãšãããšã«ããïŒ$m=14$ ãè¡šé¡ã«ããçŽ æ°é¥ã«ãããïŒå®éšå€ã $\alpha\in(\mathbb{Z}/m\mathbb{Z})^{*}$ ããšã« 1æã®ã°ã©
ãã«ããïŒã°ã©ãã®æšªè»žã¯åžžçšå¯Ÿæ°ç®ç (æ¡æ°) ãšãïŒçžŠè»žã¯ååžãšããïŒ$\beta\in(\mathbb{Z}/m\mathbb{Z})^{*}$ ããšã«è²å
ããå€ãïŒ$x$ ãã $xâ$ ã®ç¯å²ã®çŽ æ°ã«ã€ããŠæ±ããååž $(\varpi_{\alpha,\beta}(x;xâ))$ ã暪軞 $(\log_{10}(xâ)+\log_{10}(x))/2$
ã®äœçœ®ã«ç¹ã眮ãïŒååžã®æ¥µéãšããŠæåŸ ããã $1/\varphi(m)$ ã«æšªè»žã眮ãïŒ$\log_{10}(x)=6(x=10^{6})$ ã«çžŠè»žã眮ãïŒ
次ã®ã°ã©ãã¯ïŒ$m=4$ ã®ãã®ã§ããïŒèšå·ãç ©éã«ãªãã®ã§ïŒ$\varpi_{\alpha,\beta}$ ã $\mu_{p}(n;\alphaarrow\beta)$ ãªã®ã§ $\alpha,$
$\beta\in(\mathbb{Z}/m\mathbb{Z})^{*}$ ã®è¡šèšã«æ³ (mod ãªã©) ãç¥ãïŒ1 $mod m$ ãåã« 1ãšã $\varpi_{1,1},$ $\mu_{p}(n;1arrow 1)$ ãªã©ãšè¡š
ãããšã«ããïŒã $1$ modulo $4$ã 㯠$\alpha=1mod 4$ ã§ïŒå°ãå€å¥ãã«ãããïŒæšªè»žäžæ¹ã®èãè²ã®ç¹ã®åã $\varpi_{1,3}$ ã®å®éšå€ïŒäžæ¹ã®æ¿ãè²ã®ç¹ã®åã $\varpi_{1,1}$ ã®å®éšå€ã§ããïŒã$3$ modulo $4$ã ã¯ïŒæšªè»žäžæ¹ã®èãè²ã®ç¹ã®åã $\varpi_{3,3}$ ã®å®éšå€ïŒäžæ¹ã®æ¿ãè²ã®ç¹ã®åã $\varpi_{3,1}$ ã®å®éšå€ã§ããïŒ
lmodulo 4 3modulo 4
$l$. â
$\alpha=1$ ãš $\alpha=3$ ã®ã°ã©ãã¯ã»ãšãã©åãã«èŠããïŒãã㯠$\varpi_{1,1}\fallingdotseq\varpi_{3,3},$ $\varpi_{1,3}\fallingdotseq\varpi_{3,1}$ ãæå³ããïŒ\S 3ã®å®çã®èšŒæããŸã㊠$\mu_{p}(n;1arrow 1)\sim\mu_{p}(n;3arrow 3),$ $\mu_{p}(n;1arrow 3)\sim\mu_{p}(n;3arrow 1)$ ã瀺ããã®ã§ïŒã°ã©ãã®æŠåœ¢ã䌌ãŠããã®ã¯åœç¶ã§ããïŒãŸãïŒã°ã©ãã®äžã«æ³ 4㧠1ãšååãªçŽ æ°ã®å²å $\pi_{1}(x)/\pi(x)$
ãš 3ãšååãªçŽ æ°ã®å²å $\pi_{3}(x)/\pi(x)$ ã眮ããšæšªè»ž (1/2) äžã«äžŠã¶ïŒäž»èŠ³ã®åé¡ã§ã¯ãããïŒååæ¡ä»¶ä»ãçŽ æ°ååžã¯æå€ã«å€§ããªåããããïŒãã®å®éšããŒã¿ããã ãã§ãïŒçŽ æ°ã®åã¯ã³ã€ã³æã (æ³ 4ã«é¢ããå°äœ 1,3ãã³ã€ã³ã®è¡šè£ãšèŠãŠ) ã«ã¯äœ¿ããªãïŒæ¬¡ã« $m=8$ ã®ããŒã¿ãã°ã©ãã§èŠèŠåããïŒ$m=4$ ã®å¯Ÿç§°æ§ $(\varpi_{1,1}\fallingdotseq\varpi_{3,3}, \varpi_{1,3}\fallingdotseq\varpi_{3,1})$ ã®æ§ã«ïŒ
$m=8$ ãå¯Ÿç§°æ§ $\varpi_{\alpha,\beta}=.\varpi_{\alpha+c,\beta+c}$ ( $c$ ã¯å¶æ°) ããã€ã®ã§ïŒ$\alpha=1$ ã®ã°ã©ãã ãã§ååã§ããïŒ
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ãããå°ãèŠé£ããïŒäžãã $\beta$ ã 7, 3, 5, 1ã®é ã«ååžã®å€ã䞊ã³ïŒ$\beta=7$ ãš 3ã¯æ®ã©åãå€ã§ïŒã°ã©ãã§ã¯ã»ãŒéãªã£ãŠèŠããïŒ$\beta=1$ 以å€ã¯å¹³åå€ 1/4ããäžã«ããïŒ
$m=6$ ããŸãå¯Ÿç§°æ§ $(\varpi_{1,1}\fallingdotseq\varpi_{5,5}, \varpi_{1,5}\fallingdotseq\varpi_{5,1})$ ããã€ã®ã§ïŒ$\alpha=1$ ã®ã°ã©ãã ãã§ååã§ããïŒäžåŽã®èãè²ã®ç¹ã $\varpi_{1,5}$ ã§äžåŽã®æ¿ãè²ã®ç¹ã $\varpi_{1,1}$ ã§ããïŒããã蚌æã§ããããšã§ãããïŒãããŸã§èŠããšãã㧠$\beta=\alpha$ ã® $\varpi_{\alpha,\alpha}$ ãæãå°ããå€ããšãïŒ
1 modulo 4
$m=4$ ã®ã°ã©ããšãã䌌ãŠããïŒå¯Ÿæ¯ã®ããã«æšªã«äžŠã¹ããïŒèŠåãã¯é£ããïŒéããŠã¿ããšïŒ$m=6$
ã®æ¹ããããã« $m=4$ ãããå€åŽ (1/2ã«ãã暪軞ããèŠãŠ) ã«ããïŒä»¥äžïŒ$m=10,14,30$ ã®ããŒã¿ãèŠãããïŒãããŸã§ã®æ§ãªåçŽãªå¯Ÿç§°æ§ã¯ãªãïŒçŽ°ããæ°å€ã䞊
ã¹ãããšãããïŒçŽæçãããããªããïŒååžã®æšç§»ã®æŠåœ¢ãéèŠãªã®ã§å°ãå°ãããããããªãããã¹ãŠã®ã°ã©ãã䞊ã¹ãïŒ
$imd\mathfrak{v}1\circ 10$
ã$3\dot{l5}|.$
$0A|_{\mu-\sim}\cdots\cdots\cdots\cdots\cdots\cdot$
$11\cdot\circ d\mathfrak{u}1\circ 1.$
$0.\varphi|\cdot\sim.\sim\ldots\ldots$
â$\sim$
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$1, \infty ulo\cdot 0$
$\alpha=lmod l0$ ã¯äžããé ã« $\beta=7,3,9,1$ ã®ååžã䞊ã³ïŒ$\beta=7,3$ ã¯æ®ã©åãå€ããšãïŒ$\alpha=3$ mod10ã§ã¯äžããé ã« $\beta=9,7,1,3$ ãšäžŠã³ïŒ$\alpha=7mod l0$ ã§ã¯äžããé ã« $\beta=9,3,1,7$ ãšäžŠã³ïŒ$\alpha=9$
$mod 10$ ã§ã¯äžããé ã« $\beta=1,3,7,9$ ãšäžŠã¶ïŒ$m=14,30$ ã«ã€ããŠãïŒã°ã©ãã«ãããç¹ã®ééãçºããã°ïŒ$\alpha$ ããšã«ç°ãªãããšãèŠãŠãšããïŒååæ¡ä»¶ä»ãçŽ æ°ååžç¢ºç $\mu_{p}(n;\alphaarrow\beta)(=\sum_{i\geq 1}\mu_{p}(n|gap(n)=d_{i}))$ ã®äºæž¬å€ãšããŠïŒæ¬¡ã®ãã®ãèããïŒ
$\mu_{p}â(n;\alphaarrow\beta)=\sum_{i\geq 1}\mu_{p}â(n|gap(n)=d_{i})$
$\mu_{p}â(n|$ gap$(n)=d_{i})= \mu_{p}â(n, d_{i})\prod_{t=1}^{d_{l}-1}(1-\mu_{p}â(n, t))$
$\mu_{p}â(n, t)=r_{t}c_{t}\mu(n+t)$ $(ãã ã gcd(n+t, m)\neq 1$ ã®ãšã $\mu_{p}(n, t)=0$ ãšãã $)$
$c_{t}$ 㯠Hardy-Littlewood ã®äºæ³ã«çŸããå®æ°ã§ïŒ$r_{t}= \prod_{3\leq p|gcd(m,t)}(p-1)/(p-2)$ ãšããïŒHardy-Littlewood ã®äºæ³ããïŒ$n$ ãçŽ æ°ã§ãããšã®ä»®å®ã®ããšã§ $n+t$ ãçŽ æ°ã§ãã確çãšã㊠$c_{t}\mu(n+t)$
ããšããïŒä»ïŒ$n$ ã«ååæ¡ä»¶ $(n\in\alpha)$ ãä»®å®ãããã®ã§ïŒååé¡ $\alpha$ ã«å¯Ÿã㊠$t$ ã«ãã£ãŠã¯ $n+t$ ã
$m$ ãšçŽ ã§ãªãããšã«ãªãïŒãããïŒéé $t$ ã®æ¹ããçºãããŠã¿ãïŒééã $t$ ã®é£ãåã£ãçŽ æ°ã®çµ $(n, n+t)$ ã«ã€ããŠïŒ$n$ ã®å±ãããæ¢çŽå°äœé¡ã®åæ°ã $k_{d,m}$ ãšããïŒ$\varphi(m)/k_{t,m}$ ã®å€ãèšç®ãããã®ãäžã® $r_{t}$ ã§ããïŒæ³ $m$ ã«é¢ããæ¢çŽå°äœé¡ $(\varphi(m)$ å $)$ ã«åçã«ååžããçŽ æ°ã«å¯ŸããŠïŒçŽ æ°ã®çµ $(n, n+t)$ 㯠$k_{t,m}$ åã®å°äœé¡ã«åã£ãŠååžããã®ã§ïŒç¢ºç㯠$\varphi(m)/k_{t,m}(=r_{t})$ åãããã¹
ãã§ããïŒããããŠïŒååé¡ $\alpha$ ã«å±ããçŽ æ° $n$ ã«å¯ŸããŠïŒ$n+t$ ãçŽ æ°ãšãªãæ¡ä»¶ä»ã確çãšããŠ$\mu_{p}â(n, t)=r_{t}c_{t}\mu(n+t)$ ãèããïŒèªç¶æ°ãçŽ æ°ã§ããäºè±¡ã®çŽ æŽãªæå³ã§ã®ç¬ç«æ§ã¯ä¿éãããªããïŒååæ¡ä»¶ã«ãã食㚠Hardy-Littlewood ã®äºæ³ãçã蟌ãã $\mu_{p}â(n, t)$ ã«ã€ããŠã¯ïŒè¿äŒŒçãªæå³ã§ç¬ç«æ§ãä»®å®ããŠããããšèããïŒååé¡ $\alpha$ ã«å±ããçŽ æ° $n$ ã«å¯ŸããŠïŒga$P$ $(n)=d$ ãšãªã
æ¡ä»¶ä»ã確çã®äºæž¬å€ãšã㊠$\mu_{p}â(n|$ gap$(n)=d_{i})$ ããšãïŒæåäºè±¡ã®åãšããŠïŒååé¡ $\alpha$ ã«å±ãã
çŽ æ°ã«å¯ŸããŠïŒæ¬¡ã®çŽ æ°ã $\beta$ ã«å±ããæ¡ä»¶ä»ã確çãšã㊠$\mu_{p}â(n;\alphaarrow\beta)$ ãèããïŒ
$m=4,8,6$ ã§èŠã $\varpi_{\alpha,\beta}$ ã®å¯Ÿç§°æ§ãšåã察称æ§ãïŒæ¡ä»¶ä»ã確ç $\mu_{p}â(n;\alphaarrow\beta)$ ã«ãããïŒ$\alpha=1$㧠$\varpi_{1,\beta}(x;xâ),$ $\mu_{p}â(x;1arrow\beta)$ ãã°ã©ãã§èŠ³å¯ããïŒãã ãïŒ$\mu(x)=1/\log(x)$ ãšããïŒ
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ã°ã©ãã«ãããŠïŒå®éšå€ $\varpi_{1,\beta}$ ãšäºæž¬å€ $\mu_{p}â(x;1arrow\beta)$ ã§åãè²åãã«ããŠããïŒæ¬¡ã« $m=6$ ãèŠããïŒå·Šã®ãã®ã¯äžèšäºæž¬å€ãéãããã®ã§ïŒå³ã®ãã®ã¯æ¯èŒãšã㊠$\mu_{p}â(n, t)$ ã®
å®çŸ©ã§ $r_{t}$ ã®é ãæã㊠$\mu_{p}â(n, t)=c_{t}\mu(n+t)$ ã§èšç®ãããã®ã§ããïŒ
äžã§å®çŸ©ããäºæž¬å€ã¯å®éšããŒã¿ãããè¿äŒŒããŠããããã«èŠãããïŒ$r_{t}$ ã®é ã®ãªãå³ã®ãã®ã¯å®éšå€ãšããé¢ããŠããïŒå®éšã«ãã㊠$r_{t}$ ã®é ã®å¹æãè¡šããŠããïŒ
以äžïŒ$m=10,14,30$ ã«ã€ããŠïŒäºæž¬å€ãéããã°ã©ãã䞊ã¹ãïŒåé·ã«ãªããïŒå šãŠäžŠã¹ãïŒ$1$ modulo 10
7 modulo 10
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5 modulo 14
13 modulo 14
1 modulo 30
11 modulo 30
17 modulo 30 19 modulo 30
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29 modulo 30
è²åãã®å·®ã埮åŠã§ïŒãšãã©ãã°ã©ããéãªã£ãŠãããšãããããïŒèŠé£ãããšãã®äžãªããïŒããã§ãå®éšããŒã¿ã«å¯ŸããŠéåžžã«è¯ãè¿äŒŒãäžããŠããããšãèŠãŠåãããšæãïŒäºæž¬å€ $\mu_{p}â(n;\alphaarrow\beta)$
ãããå°ãç°¡åãªåŒã«çœ®ãæããããšãã§ãããïŒäºæž¬å€ã®æå³ãæ確ã«ããããã«äžã®å®çŸ©ãæ¡çšããïŒããã§ã¯ãã®ç°¡æåã«ã¯è§ŠããªãïŒä»¥äžã®å®éšãã次ãäºæ³ãããïŒ
äºæ³ 5.1 $\mu_{p}(n;\alphaarrow\beta)\sim\mu_{p}â(n;\alphaarrow\beta)$ å³ã¡ïŒ$\pi_{\alpha,\beta}(x)\sim\int_{2}^{x}\mu_{p}â(x;\alphaarrow\beta)dx$
äºæž¬å€ã«å¯ŸããŠã¯ïŒ$\mu(x)\sim 1/\log(x)$ (çŽ æ°å®ç) ãããã®æ¥µéãèšç®ã§ããïŒèšç® (蚌æ) ã¯ç ©éãªã®ã§ããã§ã¯çç¥ããïŒ
å®ç 5.2 $\mu_{p}â(n;\alphaarrow\beta)arrow\frac{1}{\varphi(m)}$ $(narrow\infty)$
$\mu_{p}â(n;\alphaarrow\beta)$ ã¯æ¡ä»¶ä»ã確ç $\mu_{p}(n;\alphaarrow\beta)$ ã®äºæž¬å€ãšããŠå°å ¥ããïŒ$\alpha$ ã«å±ããçŽ æ° $n$ ã«ã€ã
㊠$np(n)\in\beta$ ãšãªã確çã§ïŒçŽ æ° $n$ ã«å¯Ÿã㊠$n\in\alpha$ ã〠np $(n)\in\beta$ ãšãªã確çãšç°ãªãããšã泚æ
ããŠããïŒå®ç 5.2ã¯ïŒçŽ æ° $n\in\alpha$ ãåå倧ãããšããšïŒnp$(n)\in\beta$ ãšãªã確ç㯠$\alpha$ ã«äŸããªãããšãæå³ããïŒ
å®ç 5.2㯠$\alpha,$$\beta$ ãå ±ã«æ³ $m$ ã®å°äœé¡ãšããã®ã ãïŒäžè¬ã« $\alpha\in(\mathbb{Z}/m\mathbb{Z})^{*},$ $\beta\in(\mathbb{Z}/mâ\mathbb{Z})^{*}$ ãšã
ããšã $\mu_{p}â(n;\alphaarrow\beta)arrow 1/\varphi(mâ)$ ãšãªãïŒãã ãïŒ$\mu_{p}â(n, t)$ ã«ããã $r_{t}$ 㯠$\alpha$ ã®æ³ã® $m$ ã«å¯ŸããŠå®
矩ããããã®ããšãïŒå€§æ°ã®æ³åãšããŠåå倧ããçŽ æ° $n$ ã«ã€ããŠïŒnp$(n)$ ãã©ã®å°äœé¡ã«å±ããã㯠$n$ ã®ååé¢ä¿ã«ã¯äŸããªãïŒã€ãŸãïŒçŽ æ° $n$ ãšæ¬¡ã®çŽ æ° np$(n)$ ã®ããããã«ã€ããŠååæ¡ä»¶
ã§èšè¿°ãããäºè±¡ã¯ç¬ç«ã§ããïŒåå倧ãããã¹ãŠã®çŽ æ°ã«ã€ããŠïŒåååŒã§èšè¿°ãããäºè±¡ã¯ç¬ç«ã§ããïŒ
Hardy-Littlewood ã®äºæ³ãšïŒäºæ³ 5.1, äºæž¬å€ãå®çŸ©ããã«ããã£ãŠèããæšè«ããã¹ãŠèªãããïŒé©åœãªä»£çšç©ã«çœ®ãæããŠãã®æç«ãèªããããšãã§ãããªãã°ïŒå®ç 5.2ãã次ãåŸãïŒ
ç³» 5.3 $\alpha\in(\mathbb{Z}/m\mathbb{Z})^{*},$ $\beta\in(\mathbb{Z}/mâ\mathbb{Z})^{*}$ ã«å¯ŸããŠïŒ$\lim_{xarrow\infty}\frac{\pi_{\alpha,\beta}(x)}{\pi_{\alpha}(x)}=\frac{1}{\varphi(m)}$
\S 6. çµã³
æ¡ä»¶ä»ã確çã®äºæž¬å€ $\mu_{p}â(n;\alphaarrow\beta)$ ã¯ïŒæçãªæ£é çŽæ°ãªã®ã§ãšãããåæããŸãïŒæ¥µããŠåæã®é ãçŽæ°ãªã®ã§ïŒæå³ã®ããæ°å€ãèšç®ããã«ã¯ç·åæ³ãããŸã工倫ããªããã°ãªããŸããïŒè©³ãã説æã¯çããŸããïŒåæãé ãããïŒååæ¡ä»¶ä»ãçŽ æ°ååž $\pi_{\alpha,\beta}(x)/\pi_{\alpha}(x)$ ã®æ¥µéå€ã¯èšç®ã§ããŸããã (ç³» 5.3), 誀差é ã®è©äŸ¡ã¯ã§ããŸããã§ããïŒèª€å·®é ãååæ¡ä»¶ä»ãçŽ æ°ååžã®åããèšè¿°ããã®ã§ïŒè©äŸ¡ã欲ãããšããã§ãïŒçŸæç¹ã§ã¯å®ç 3.1,3.2ã®ç¹å¥ãªå Žåã ãã§ïŒãã®å Žåã¯è¿äŒŒãšèšãããçå·ãšèŠãªããŠãããããã誀差é ãå°ãããªã£ãŠããŸãïŒäžè¬ã®å Žåã¯èª€å·®é
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ãã¡ãã£ãšå€§ããã§ïŒæ°åæ¡çšåºŠã®çŽ æ°ã«å¯ŸããŠãååžã«åããæ®ã£ãŠããŸãïŒçŽ æ°ééã®è©äŸ¡ã䜿ã£ã確ççè©äŸ¡ãã§ããŸããïŒãŸã å°ãæ確ãã«æ¬ ããŠããïŒç 究éäžã«ãããŸãïŒååæ¡ä»¶ä»ãçŽ æ°ååžãçæ³ããåæ©ãšãªã£ãïŒå€§å¡åïŒæ¥é«åã®å®éšèŠ³å¯ãã $g_{d}(x)/\pi(x)$ ã¯ïŒ
é£ç¶ãã 3ã€ã®çŽ æ°ã«å¯Ÿããååé¢ä¿ã«é¢ããçŽ æ°ååžã§ãïŒåå倧ããçŽ æ°ã«å¯ŸããŠåååŒã§èšè¿°ãããäºè±¡ãè¿äŒŒçã«ç¬ç«ã§ããããš (å®ç 5.2, Hardy-Littlewood ã®äºæ³ãªã©ãä»®å®ããŠãã $)$ ãèªããã°ïŒ$\tilde{g}_{d}(x)(=\#\{p\leq x|d|d_{p}\}=\sum_{d|dâ}g_{dâ}(x),$ $dâ>2x$ 㧠$g_{dâ}(x)=0$ ãªã®ã§æéå $)$ ã«ã€
ã㊠$\lim_{xarrow\infty}\tilde{g}_{d}(x)/\pi(x)=1/\varphi(d)^{2}$ ãšãªãïŒ$g_{d}(x)= \sum_{k\geq 1}\mu(k)\tilde{g}_{kd}(x)(kd>2x$ 㧠$\tilde{g}_{kd}(x)=0$ ãª
ã®ã§æéå) ãªã®ã§ïŒ
$\lim_{xarrow\infty}\frac{g_{d}(x)}{\pi(x)}=\lim_{xarrow\infty}\sum_{k\geq 1}\mu(k)\frac{\tilde{g}_{kd}(x)}{\pi(x)}=\sum_{k\geq 1}\mu(k)\lim_{xarrow\infty}\frac{\tilde{g}_{kd}(x)}{\pi(x)}=\sum_{k\geq 1}\frac{\mu(k)}{\varphi(kd)^{2}}$
$g_{d}(x)/\pi(x)$ ã®å€ã幟ã€ãèšç®ãè¡šã«ãŸãšããïŒè¿äŒŒå€ã¯æçµæ¡ã§åãæšãŠãšããïŒ
åååŒã§èšè¿°ãããäºè±¡ã®ç¬ç«æ§ãä»®å®ããã«ã¯ 8æ¡ã 9æ¡çšåºŠã®çŽ æ°ã§ã¯å°ããã®ã§ïŒäºæž¬æ¥µéå€ãšã®å·®ã倧ãããŠãä»æ¹ãªãããšãªã®ãããããŸããïŒç¬ç«æ§ãå®çŽã«ä»®å®ããŠããããšã«åå ãããã®ãããããŸããïŒåã«åæãé ãã ããããããŸããïŒèª€å·®é ãè©äŸ¡ã§ããŠããªãããïŒå€§å¡åïŒæ¥é«åã®å®éšèŠ³å¯ã«å¯Ÿããå®éçäºæž¬ãåŸãããŠããŸããïŒåœŒãã®åãã«çããããŠããŸããïŒ
æ°çç ã®ç 究éäŒã§ã¯ïŒHardy-Littlewood ã®äºæ³ã®ããšã§ã説æã§ããŠããªããšçµã³ãŸããïŒãã®åŸã®ç 究ã§ïŒå¹Ÿã€ãä»®å®ããããŸããã Hardy-Littlewood ã®äºæ³ã®ããšã§èª¬æã§ããããšãããããŸããïŒç 究éäžã®äžå®å šãªãã®ãæ瀺ããŠããŸãïŒç³ãèš³ãããŸããã§ããïŒæåŸã«ãã®ãããªå®éšèŠ³å¯ã«çºè¡šã®æ©äŒãäžããŠãã ãããŸããããšïŒç 究éäŒç 究代衚è ç°äžåææ°ã«æè¬ããããŸãïŒ
References
[1] R. Crandall-C. Pomerance, Prime Numbers -$A$ Computational Perspective, Second Edition,Springer-Verlag Berlin-Heidelberg-New York, 2005.
[2] R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer-Verlag Berlin-Heidelberg-New York, 2004.
[3] G. Hardy, Collected Works of G. H. Hardy, Clarendon Press, Oxford, 1966.
[4] G. H. Hardy-E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., ClarendonPress, Oxford, 1979.
[5] æŸæ¬èäºïŒãªãŒãã³ã®ãŒãŒã¿é¢æ°ïŒæåæžåºïŒ2005.
[6] æ¬æ©æŽäžïŒè§£æçæŽæ°è« IäžçŽ æ°ååžè«äžïŒæåæžåºïŒ2009.
[7] W. Narkiewicz, The Development of Prime Number Theory, Springer-Verlag Berlin-Heidelberg-New York, 2000.
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[8] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge Uni-versity Press, 1995.
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