162 $\cdots$ 2, 3, 5, 7, 11, 13, ( deterministic ) $\mathbb{r}$ ( -1 3 ) ( ) $\text{ }$ ( ). straightforward ( ) $p$ version ( ) - 2 $\mathrm{n}$ $\om

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1 $L$ (Hirofumi Nagoshi) Research Institute for Mathematical Sciences, Kyoto Univ. 1. $L$ ( ) 2. ( 0 1 ) $X_{1},$ $X_{2},$ $X_{3},$ $\cdots$ $n^{-1/2}(x_{1}+$ $X_{2}+\cdots+X_{n})$ $X_{i}$ ( ) ( ) Probabilisitic number theory ( deterministic )

$162 $\cdots$ 2, 3, 5, 7, 11, 13, ( deterministic ) $\mathbb{r}$ ( -1 3 ) ( ) $\text{ }$ ( ).$

2 162 $\cdots$ 2, 3, 5, 7, 11, 13, ( deterministic ) $\mathbb{r}$ ( -1 3 ) ( ) $\text{ }$ ( ). straightforward ( ) $p$ version ( ) - 2 $\mathrm{n}$ $\omega(n)$ $n$ $p$ $\omega(n)$ $n$ 1 Erd\"os Kac [EK1][EK2] $\omega(n)$

3 $\mathrm{l}$ 163 ( $\mathrm{c}\mathrm{a}\mathrm{n}$ Kac one hear the $\mathrm{d}\mathrm{r}\mathrm{u}\mathrm{m}?\rfloor$ shape of a Feynman-Kac ) Kac Theorem 2.1. $x_{1},$ $x_{2}\in \mathbb{r}(x_{1}<x_{2})$ L $\lim_{narrow\infty}\frac{1}{n}\neq\{1\leq n\leq N x_{1}<\frac{\omega(n)-1\mathrm{o}\mathrm{g}1\mathrm{o}\mathrm{g}n}{\sqrt{1\mathrm{o}\mathrm{g}1\mathrm{o}\mathrm{g}n}}<x_{2}\}=\frac{1}{\sqrt{2\pi}}\int_{x_{1}}^{x_{2}}e^{-\frac{x^{2}}{2}}dx$ 2 ${\rm Re}(s)=1/2$ Selberg [S3] [S1] Theorem 2.2. $\mathbb{c}$ $E$ $\lim_{tarrow\infty}\frac{1}{t}m(\{t\in[0, T]$ $\frac{1\mathrm{o}\mathrm{g}\zeta(\frac{\mathrm{l}}{2}+it)}{\sqrt{\frac{1}{2}1\mathrm{o}\mathrm{g}\log t}}\in E\})=\frac{1}{2\pi}\int\int_{E}e^{-\frac{x^{2}+}{2}u_{-}^{2}}$ dxdy $\mathrm{r}$ $m$ $\log\zeta(s)$ ( ${\rm Re}(s)arrow\infty$ 0[ Selberg class [S3] [BH] [BH] 3. Selberg $L$ ( ) $\{s\in \mathbb{c} 0\leq{\rm Re}(s)\leq 1\}$ ( ${\rm Re}(s)=0,1$ ) : 1 2

$Goldbach 164 1 $\pi(x)\sim x/\log x$ ${\rm Re}(s)=1$ $\zeta(s)\neq 0$ on ${\rm Re}(s)=1$ Hadamard de la Valli Poussin I 3 1 2 Eisenstein ${\rm Re}(s)=1$ $\zeta(s)\neq 0$ 3 2 Selberg Erdi Sieve$

4 Goldbach $\pi(x)\sim x/\log x$ ${\rm Re}(s)=1$ $\zeta(s)\neq 0$ on ${\rm Re}(s)=1$ Hadamard de la Valli Poussin I Eisenstein ${\rm Re}(s)=1$ $\zeta(s)\neq 0$ 3 2 Selberg Erdi Sieve (Elementary method) - Hardy-Littlewood prime -tuple conjecture. $n$ $p_{n+1}-p_{n}$ ( $p_{n}$ $n$ ) $\text{ }x_{n}:=p_{n}/\log p_{n}$ $x_{n+1}-x_{n}$ ( ) $\text{ }$ ( Sieve Method, Elementary Method) 2 (see [La]) Theorem 3.1. $D=\{s\in \mathbb{c} 1/2<{\rm Re}(s)<1\}$ $K$ $K$ $K$ $h(s)$ $\epsilon>0$ $t\in \mathrm{r}$ [ $\sup_{s\in K} \zeta(s+it)-h(s) <\epsilon$ $\lim_{tarrow}\inf_{\infty}\frac{1}{t}m(t\in[0,t] \sup_{s\in K} \zeta(s+it)-h(s) <\epsilon)>0$

$$\psi\mathrm{a}$ 165 $D$ - $\log$

5 $\psi\mathrm{a}$ 165 $D$ - $\log p$ $\mathbb{q}$ ( ) 1, 2 $\mathrm{v}$ $\sum_{n}n^{-s}$ ${\rm Re}(s)>1$ ( ) $0<{\rm Re}(s)<1$ Sieve 4. $f\mathrm{j}$ Wigner ( 1 ) (see e.g. $)$ [M] ( GUE(Gaussian Unitary Ensemble) J $\mathcal{h}_{n}$ $N\cross N$ -Hermite Gauss $P_{N}(dX)\propto$ $\exp(-\mathrm{r}(x^{2}))dx$ $(\mathcal{h}_{n}, P_{N})$ GUE. $Narrow\infty$ Montgomery (1973)

6 166 2 $\rho_{i}$ $\gamma_{i}$ Dyson GUE Odlyzko (1987) $n$ [KS] [C] survey GUE GUE GUE Dyson CUE(Circular Unitary Ensemble) $N\cross N$- $U(N)$ Haar $Q_{N}$ $(U(N), Q_{N})$ scaled limit $Narrow\infty$ $M_{k}(T)= \int_{0}^{t} \zeta(1/2+it) ^{2k}dt$ ) CUE $Z(U, \theta)=\det(i-ue^{-i\theta}),$ $U\in U(N)$ $k=1,2$ $\lim_{narrow\infty}n^{-k^{2}}q_{n}( Z(U, \theta) ^{2k})$ $k$ $M_{k}(T)$ ${\rm Re}(s)=1/2$ Keating-Snaith(2000) Theorem 4.1. $\mathbb{c}$ $E$ $\lim_{narrow\infty}q_{n}(u\in U(N)$ $\frac{1\mathrm{o}\mathrm{g}z(u,\theta)}{\sqrt{\frac{1}{2}1\mathrm{o}\mathrm{g}n}}\in E)=\frac{1}{2\pi}\int\int_{E}e^{-_{2}^{\underline{x^{2}}+x_{-}^{2}}}dxdy$ ( $\theta$ ) Selberg Theorem 2.2 \cdot$ }$\backslash 5.

7 $\text{ }-$ $\mathbb{c}$ 167 Theorem 22 ( ( $t(={\rm Im}(s))$ $L$ $t$ $\chi \mathrm{m}\mathrm{o}\mathrm{d} q$ $L$ $L(s, \chi)$ ( $q$ $t$ $q$ [S2] $t$ $L$ $N$ $k$ [ILS] 3 $L$ [N1] $t$ $t$ Theorem 22 $L$ $N$ $\mathcal{f}_{n}$ $\Gamma_{0}(N)(\subset SL(2, \mathbb{z}))$ 2 Hecke eigen cusp forms $N$ $f\in \mathcal{f}_{n}$ Hecke $\lambda_{f}(n)$ $T_{n}(N)$ $n$ Hecke $T_{n} (N)f=\lambda f(n)f$, where $T_{n} (N):=T_{n}(N)/n^{\frac{1}{2}}$ $L$ $(s, f):= \sum_{n=1}^{\infty}\frac{\lambda_{f}(n)}{n^{s}}=\prod_{p N}(1-\frac{\lambda_{f}(p)}{p^{s}})^{-1}$ $(1- \frac{\lambda_{f}(p)}{p^{s}}+\frac{1}{p^{2s}})^{-1}$ for ${\rm Re}(s)>1$ Hecke ${\rm Re}(s)=1/2$ $N$ $N$ Theorem 5.1. $t\neq 0$ $\mathrm{r}$ $E$ $\lim_{narrow\infty}\frac{1}{\# F_{N}}\#\{f\in F_{N}$ $\frac{{\rm Im}\log L(\frac{1}{2}+it,f)}{\sqrt{\frac{1}{2}\log 1\mathrm{o}\mathrm{g}N}}\in E\}=\frac{1}{\sqrt{2\pi}}\int_{E}e^{-\frac{x^{2}}{2}}dx$.

8 168 $N$ Hecke eigen cusp forms $f\in \mathcal{f}_{n}$ $Narrow\infty$ $f$ $\mathrm{r}$ ) 6. [N2] Moment method Theorem 6.1. $m\in- \mathrm{n}$ $t\neq 0$ $\sum_{f\in F_{N}}({\rm Im}\log L(\frac{1}{2}+it,$ $f))^{m}$ $=C_{m} \#\mathcal{f}_{n}(\frac{1}{2}\log\log N)^{\frac{m}{2}}+O_{m,t}(N(\log\log N)^{\frac{m-1}{2}})$. $C_{m}=\{\begin{array}{l}m!!2T,ifmiseven0,ifmisodd\end{array}$ $C_{m}$ Theorem Selberg [S1] explicit formula Lemma 6.2. ${\rm Re} s\geq 1/2_{\text{ }}x\geq 10$ $\frac{l }{L}(s, f)=-\sum_{n\leq x^{3}}\frac{\lambda_{x}(n)c_{f}(n)}{n^{s}}+\frac{1}{\log^{2}x}\sum_{\rho}\frac{x^{\rho-s}(1-x^{\rho-s})^{2}}{(s-\sqrt)^{3}}$ (6.1) $- \frac{1}{\log^{2}x}\sum_{l=0}^{\infty}\frac{x^{-\frac{1}{2}-\ell-s}(1-x^{-\frac{1}{2}-\ell-s})^{2}}{(s+\frac{1}{2}+\ell)^{3}}$.

9 169 $\rho$ $\ovalbox{\tt\small REJECT}(n)$ ( $L(s, f)$ $\ovalbox{\tt\small REJECT}_{x}(n)$ ( von Mangolt where $\ovalbox{\tt\small REJECT}_{x}(n)\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}(n)w_{x}(n)$ $w_{x}(n):=-\{_{0}^{1}\{$ $, \frac{1}{2}\log 2\frac{x^{3}}{n}-\log^{2}\frac{x^{2}}{n})/\log^{2}x$ for $1\leq n\leq x$,, for $x<n\leq x^{2}$, $\frac{1}{2}\log 2\frac{x^{3}}{n})/\log^{2}x$, for $x^{2}<n\leq x^{3}$, for $x^{3}<n$. (6.1) $\sum_{\rho}$ ( $\sum_{\ell}$ ) (by Kowalski-Michel1999) Lemma 6.3. $N$ $A>0$ : $t_{1},$ $t_{2}$ with $t_{1}<t_{2}$, $t_{2}-t_{1} \geq\frac{1}{\log q}$ $\alpha\geq 1/2+(\log N)^{-1}$ $c$ with $0<c<1/4$ ( $\sum_{f\in F_{N}}N_{f}(\alpha, t_{1}, t_{2})<<_{c}(1+ t_{1} + t_{2} )^{A}N^{1-c(\alpha-\frac{1}{2})_{(\log N)(t_{2}-t_{1})}}$ $\vee\supset_{\text{ }}$ $arrow>$. $N_{f}(\alpha, t_{1}, t_{2})l\mathrm{h}_{\text{ }}L(s, f)$ $\rho=\beta+i\gamma$ $\beta\leq\alpha$, $t_{1}\leq\gamma\leq t_{2}$. ( $\not\in-$ ) (6.1) Lemma 6.4. $N$ $(n, N)=1$ $\mathrm{t}\mathrm{r}t_{n} (N)=\frac{(N+1)}{12}n^{-1/2}\delta_{n=\square }+O(n^{c}N^{1/2})$. $c>0$ $\delta_{n=\square }$ $n$ 1 $n$ 0 ${\rm Im} \sum_{p\leq N^{\delta}\infty}p^{12+it}\lambda_{f}(p)$ $\delta$ ) Theorem 5.1, Theorem 6.1 $\log\log N$ $\sum_{p\leq x}1/p=\log\log x+o(1)$

$:Limit \mathrm{f}\mathrm{u}\mathrm{n}\overline{\mathrm{c}}$ tions $\mathrm{v}^{\mathrm{a}}$ $\mathrm{v}$ 170 7.$

10 :Limit \mathrm{f}\mathrm{u}\mathrm{n}\overline{\mathrm{c}}$ tions $\mathrm{v}^{\mathrm{a}}$ $\mathrm{v}$ rx- deterministic [ $\nearrow^{\text{ }}$ [ 1933 (Kolmogorov ) ( ) SatO-Tate SatO-Tate ( symmetric power L-functions ) $\mathrm{v}^{\mathrm{a}}$ REFERENCES products, Duke Math. J. 80 (1995), $L- [C] J. B. Conrey: and random matrices, in Mathematioe unlimited-2001 and beyond Springer-Verlag, 2001, $\mathrm{i},$ [E] P. D. T. A. Elliott: Probabilistic number theory, $\mathrm{i}\mathrm{i}$, Springer-Verlag, 1979, [EK1] P. Erdi, M. Kac: On the Gaussia law of errors in the theory of additive functions, $20\triangleright Proc. Nat. Acad. Sci. USA. 25 (1939), 207$. [EK2] P. Erd\"os, M. Kac: The Gaussia law of errors in the theory of additive numbertheoretic functions, Amer. J. Math. 62 (1940), [ILS] H. Iwaniec, W. Luo, P. Sarnak: Low lying zeros of families of Afunctions, IHES 91 (2001), [K] M. Kac: Statistical independence in probability, analysis and number theory, Carus Monograph No. 12, [KS] N. M. Katz, P. Sarnak: Zeros of zeta functions and symmetry, Bull. Amer. Math. Soc. 36 (1999), [Ku] J. Kubilius: Probabilistic methods in the theory of numbers, AMS, $\mathrm{l}\mathrm{a}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{n}\check{\mathrm{c}}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{i}$ [La] A. Theorems for the Riemann Zeta-Function, Kluwer, [Li] U. V. Linnik: Ergodic properties of algebraic fields, Springer-Verlag, [M] M. L. Mehta: Random matrices, Academic Press, [N1] H. Nagoshi: The universality of families of automorphic&functions, preprint. distribution and afamily of automorphic preprint. [N2] H. Nagoshi: Gaussian- $\mathrm{l}$-functions,

11 Sinai: 48 \mathrm{r}\mathrm{y},$ Princeton. 171 [S1] A. Selberg: Contributions to the theory of the Riemann zeta-function, Arch$\cdot$ Math. $\cdot$ Naturvid (1946), ; Collected Papers, Springer-Verlag. $\mathrm{v}\mathrm{o}\mathrm{l}1, ,$ $\mathrm{s}\mathrm{k}\mathrm{r}$ [S2] A. Selberg: Contributions to the theory of Dirichlet s L-function, Norske Vid. Akad. Oslo (1946), 1-62: Collected Papers, Springer-Verlag. $\mathrm{v}\mathrm{o}\mathrm{l}1, ,$ [S3] A. Selberg: Old and new conjectures and results about a class of Dirichlet series, Proceedings $\mathrm{t}\mathrm{h}\infty of the Amalfi Conference on Analytic Number \mathrm{r}\mathrm{y}(1992)$, ; Collected Papers, Springer-Verlag. $\mathrm{v}\mathrm{o}\mathrm{l}2,47-63,$ $\mathrm{g}\cdot$ $\mathrm{e}\mathrm{r}\mathrm{g}6\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{h}\infty [Si] Y. Introduction to University Press, 1976.

SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T

SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary