Koyama, Shin-ya

Kyoto University Research Information Repository

TitleON THE RESULTS OF SARNAK-RUDNICK-KATZ-IWANIEC-LUO ON ZEROS OF ZETA FUNCTIONS(Number Theory and its Applications)Author(s)Koyama, Shin-yaCitation数理解析研究所講究録 (1998), 1060: 145-153Issue Date1998-08URL http://hdl.handle.net/2433/62364RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityON THE RESULTS OFSARNAK-RUDNICK-KATZ-IWANIEC-LUOON ZEROS OF ZETA FUNCTIONS(Shin-ya Koyama)C.ontents.$0$ . Introduction1. The Riemann Zeta Function (Rudnick-Sarnak 1994)2. Congruence Zeta Functions (Katz-Sarnak 1996)3. Examples of “Global Monodromy”(Katz-Sarnak 1996)( $\mathrm{I}_{\mathrm{W}}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{e}\mathrm{c}-\mathrm{L}\mathrm{u}\mathrm{o}$-Sarnak 1997)4. Applications$(\mathrm{I}\mathrm{w}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{e}\mathrm{c}-\mathrm{L}\mathrm{u}\mathrm{o}- \mathrm{s}_{\mathrm{a}}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{k} 1997)$(Iwaniec-Sarnak 1997)’$0$ . INTRODUCTIONIn this article we will survey the results of Sarnak and surrounding mathemati-cians of him on the distribution of zeros of zeta functions. The chief concern is thespacing distribution of zeros. We begin with explaining what spacing distributionis.Spacing Distribution.Let $\lambda_{0}\leq\lambda_{1}\leq\cdots$ be a sequence of real numbers. Put $N(T)=\#\{j|\lambda_{j}\leq T\}$ .For considering the spacing distribution, we can normalize $\lambda_{j}$ such that $N(T)\sim$$T$ $(Tarrow\infty)$Define the measure by$\mu(N)=\mu(\lambda_{0}, \cdots, \lambda_{N})[a, b]=\frac{1}{N}\#\{0\leq j\leq N-1|\triangle_{j}\in[a, b]\}$$(\triangle_{j}=\lambda_{j+1}-\lambda_{j})$which we call the spacing distribution.1060 1998 145-153 145Examples.(1) Prime NumbersLet $p_{j}$ be the j-th prime. By the prime number theorem, its normaliza-tion is given by $\lambda_{j}=\frac{p_{j}}{\log p_{j}}$ . Numerical experiments [KS2, Figure 1] suggestthat $\mu(N)arrow e^{-x}dx$ . But we have no way to prove it up to the present.(2) Zeros of the Riemann zeta functionFor numerical example we can assume the Riemann Hypothesis. Let$\rho_{j}=\frac{1}{2}+\gamma_{j}\sqrt{-1}$ be non-trivial zeros with $0<\gamma_{1}\leq\gamma_{2}\leq\cdots$ . The normal-ization is given by $\tilde{\gamma}_{j}=\underline{\gamma_{j}\log\gamma_{j}}$ by Riemann. The well-known numerical$2\pi$experiments by Odlyzko $[0]$ suggest that the spacing distribution $\mu(N)$tends to that of GUE. We call this phenomenon the Montgomery-OdlyzkoLaw.1. THE RIEMANN ZETA FUNCTIONIn this section we explain the result of Rudnick and Sarnak [RS] on the spacingdistribution of zeros of the Riemann zeta function. Although their main theoremdoes not assume the Riemann Hypothesis, here we assume it for simplicity. As wasgiven in the previous section, we have the normalized sequence $\tilde{\gamma}_{j}=\frac{\gamma_{j}\log\gamma_{j}}{2\pi}$ . Let$B_{N}=\{\tilde{\gamma}_{1}, \ldots,\tilde{\gamma}_{N}\}$ be the set of first $N$ elements. What we want to know is thenumber$N(a, b)=\#\{k|\tilde{\gamma}_{k+1^{-}\tilde{\gamma}_{k}}\in[a, b]\}$for any interval $[a, b]$ . But this quantity is hard to treat directly, since it is hardto tell if two elements are consecutive or not unless we know all members in thesequence. So we put$N_{2}=N_{2}(a, b)=\neq\{(\tilde{\gamma},\tilde{\gamma})/\in B_{N}^{2}|\tilde{\gamma}<\tilde{\gamma}’,\tilde{\gamma}’-\tilde{\gamma}\in[a, b]\}$and$N_{3}=N_{3}(a, b)=\neq\{(\tilde{\gamma},\tilde{\gamma}^{\prime/},\tilde{\gamma})/\in B_{N}^{3}|\tilde{\gamma}<\tilde{\gamma}’<\tilde{\gamma}’/,\tilde{\gamma}’’-\tilde{\gamma}\in[a, b]\}$and so on. To be precise, for a positive integer $n$ , the integer $N_{n}$ is the number of$n$-tuples whose difference between the biggest and the smallest elements belong to$[a, b]$ . Then we have inductively$N(a, b)=N_{2}-N_{3}+N_{4}$ –. . .which is a finite alternating sum. Therefore it suffices to obtain the numbers $N_{n}$for $n=2,3,4\ldots$ . Rudnick and Sarnak generalized $N_{n}$ as follows:146Definition. ($n$ -level correlation) Let $f$ be a function of $n$ -variables. The n-levelcorrelation is$R^{(n)}(f, B_{N})= \frac{1}{N}(j_{1,\mathrm{d}\mathrm{i}_{6\mathrm{t}}\mathrm{i}\mathrm{n}\mathrm{C}},.\sum_{\mathrm{t}}.,f(\tilde{\gamma}j1’\ldots,\tilde{\gamma}jn)jn)$The special case when$f(\cdots)=\{$1 $\max_{k}\tilde{\gamma}_{j_{k}}-\min_{k}\tilde{\gamma}_{j_{k}}\in[a, b]$$0$ otherwisegives$R^{(n)}(f, B_{N})= \frac{n!}{N}N_{n}$ .So the $n$-level correla.tion is a generalization of $N_{n}$ . In what follows we will study$R^{(n)}(f, B_{N})$ .For our purpose, it suffices to consider functions $f$ satisfying the following threeconditions:(1) $f(x_{1,\ldots,n}x)$ is symmetric.(2) $f(x_{1}+t, \ldots, x_{n}+t)=f(x_{1\cdots,n},x)$ for $t\in$ R.(3) $f(x)arrow \mathrm{O}$ rapidly as $|x|arrow\infty$ in the hyperplane $\sum_{j}x_{j}=0$ .Theorem 1 $(\mathrm{R}\mathrm{u}\mathrm{d}\mathrm{n}\mathrm{i}\mathrm{c}\mathrm{k}- \mathrm{S}\mathrm{a}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{k})[\mathrm{R}\mathrm{S}]$ . Assume that $f$ satisfies the above threeconditions and that its Fourier transform $\hat{f}$ satisfies$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\hat{f}(\xi))\subset\{\sum_{j}|\xi_{j}|<2\}$ ,then$R_{n}(f, B_{N}) arrow\int_{\mathrm{R}^{n}}f(x)W_{n}(X)\delta(\frac{x_{1}+\cdots+x_{n}}{n})dX1\cdots dX_{n}$as $Narrow\infty$ , where $\delta(x)$ is the Dirac mass at $\mathit{0}$ and $W_{n}(x)= \det(\frac{\sin\pi(xi-xj)}{\pi(x.-x_{j})}.)$ .Idea of Proof.$\cdot$ . By using the explicit formula, we can transform the sum in thedefinition of the $n$ -level correlation to the sum over prime numbers. It is the for-mula of the type$\mathrm{d}\mathrm{i}’ \mathrm{s}.\mathrm{t}.\mathrm{i}\sum_{(j_{1}}.\mathrm{n}’ jn^{)}\mathrm{c}\mathrm{t}=(\mathrm{p}_{1},\ldots,\mathrm{p}_{n})\sum_{\mathrm{p}_{\Gamma \mathrm{i}}\mathrm{m}\mathrm{e}\mathrm{s}}$Then we calculate the latter sum by a verycomplicated combinatorial technique. $\square$1472. CONGRUENCE ZETA FUNCTIONSIn this section we introduce the results of Katz and Sarnak [KS] [KS2]. Let $C/\mathrm{F}_{q}$be a curve with function field $k$ . The congruence zeta function is defined as$\zeta(C,$ $T)=$ $\prod$ $(1-T^{\deg()}v)^{-1}$$v$ :place of $k$It is well-known that it has the following expression.$\zeta(C, T)=\frac{P(C,T)}{(1-T)(1-qT)}$ ,where $P$ is a polynomial of degree $2g$ with $g$ being the genus. The Riemann Hy-pothesis, which was proved by Deligne, asserts all zeros of $P$ lie on $|T|=q^{-1/2}$ . Weput zeros as $e^{i\theta_{j}}q^{-1/2}(j=1,2, \ldots, 2g)$ Out interest is the spacing distribution of$\{\theta_{j}\}$ . $\mathrm{A}\mathrm{l}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\dot{\mathrm{g}}\mathrm{h}$ there are only finite number of elements, an interesting phenome-non is observed if the curve $C$ varies. We will consider a family of congruence zetafunctions and will take a certain limit in the family.By the process described before we have the measure $\mu c$ which we are interestedin. Their main theorem asserts the me.asure $\mu c$ tends to a universal one derivingfrom general classical compact groups.For a unitary matrix $A$ , we have a finite sequence of eigenvalues of it. We definethe measure $\mu_{A}$ by the same procedure from the sequence. The following lemmaassures the existence of the universal measure.Lemma (Katz-Sarnak). There exists a measure $\mu_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{V}\mathrm{e}\mathrm{r}}\mathrm{s}\mathrm{a}1\mathit{8}uch$ that$\lim_{Narrow\infty}\int_{G(N)}\mu_{A}dA=\mu$universalwith $G=U,$ $SU,$ $O,$ $so,$ $Us_{p}$ .When two measures $\mu$ and $\nu$ are given, we put $D( \mu, \nu)=\sup|\mu((-\infty, X]$ -$x\in \mathrm{R}$$\nu(-\infty, x])|$ which is called the discrepancy of the measures. If $D(\mu, \nu)$ is zero, thetwo measures are essentially equal. The following theorem says our measure $\mu_{C}$“converges” to the universal measure.Theorem 2. (Katz-Sarnak). Let $M_{g}(\mathrm{F}_{q})$ be the set of isomorphism classes ofcurve8 with genus $g$ . Then$\sum$ $D(\mu c, \mu_{unive}rsal)$$C\in M_{g}(^{\mathrm{p}_{q})}$$g arrow\infty\lim_{qarrow\infty}$$\neq M_{g}(\mathrm{F}_{q})$$=0$148Sketch of the Proof. There are following three keys in the proof.(1) The monodromy group $=\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}Sp(2g)$ .(2) Equidistribution theorem of Deligne(3) Law of large numbersWe deduce the map$M_{g}(\mathrm{F}_{q})\ni Carrow \mathrm{F}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{u}\mathrm{s}\theta(C)\in USp(2g)$is. surjective. $\square$3. EXAMPLES OF “GLOBAL MONODROMY”In the title, Global Monodromy has to be in double quotation marks, becauseit cannot be defined precisely. What we learned from the function field case isthat interesting phenomena arise when we consider a family of zeta functions. Themonodromy group plays the role of gluing those zeta functions. In the proof ofthe Riemann Hypothesis for function field cases, Deligne proved it not for a singlezeta function, but for a family of zeta functions altogether. So if the originalRiemann Hypothesis should be solved, we expect that we will be able to discover afamily of zeta functions together with its monodromy group, which dominates thezeros. Although the definition of family and monodromy is not discovered at thepresemt, we will $\overline{1}\mathrm{n}\mathrm{t}^{\mathrm{R}}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{C}\mathrm{e}$ some results which will make us believe the existence ofmonodromy groups for various families of zeta functions.Philosophy. Let $f$ be a source of $L$ -function $\mathit{8}uch$ as an automorphic form. $We$want to $con\mathit{8}ider$ a family $\mathcal{F}$ of $f$ . Although we have no precise rule on what familywe can treat, we can give some examples later. For an element $f\in \mathcal{F}_{f}$ we assumethe conductor of $f$ is $defined_{f}$ which is a positive number and $i_{\mathit{8}}$ denoted by $c_{f}$ . Put$\mathcal{F}_{X}=\{f\in \mathcal{F}|c_{f}\leq X\}$ . Let $\frac{1}{2}+i\gamma_{f}^{(j)}(j=1,2, \ldots)$ be nontrivial zeros of $L(\mathit{8}, f)$ .The Generalized Riemann Hypothesis $a\mathit{8}serts\gamma f(j)\in$ R. For zeros with $\gamma_{f}^{(j)}\in \mathrm{R}$ , we$as\mathit{8}ume\gamma_{f}^{(i)}\leq\gamma_{f}^{(j)}$ if $i\leq j$ . We put the sequence $\hat{\gamma}_{f}^{(j)}=\frac{\gamma_{f}^{(j)}\log C_{f}}{2\pi}$ The $di_{\mathit{8}}tribution$of the j-th lowest zero is defined by$\mu_{j}(X, \mathcal{F})[a, b]=\frac{1}{|\mathcal{F}_{X}|}\#\{f\in \mathcal{F}|c_{f}\leq X,\hat{\gamma}_{f}^{(j)}\in[a, b]\}$The density of low-lying zero8 in $O( \frac{1}{\log c_{f}})$ is defined by$W(x, \mathcal{F}, \phi)=\frac{1}{|\mathcal{F}_{X}|}\sum_{\mathrm{c}_{f}\leq x}D(f, \phi)$ ,149where$D(f, \phi)=\sum_{i}\phi(\hat{\gamma}_{f}^{(})j)$and $\phi i_{\mathit{8}}$ a rapidly decreasing function defined on R. (If $\gamma_{f}^{\langle j)}$ is not $real_{\rangle}$ the valueof $\phi$ should be $con\mathit{8}idered_{\mathit{0}\mathit{8}}$ zero.)We hope that$\mu_{j}(x, \mathcal{F})arrow\mu_{j}(\mathcal{F})$for $\mathit{8}ome\mu_{j}(\mathcal{F})$ and that$W(X, \mathcal{F}, \phi)arrow\int_{-\infty}^{\infty}\phi(X)W(\mathcal{F})dx$for some $W(\mathcal{F})dxa\mathit{8}Xarrow\infty$ .In the function field $ca\mathit{8}e$ we observed that $\mu_{j}(x, \mathcal{F})$ and $W(X, \mathcal{F}, \phi)$ were deter-mined by the limit of the monodromy group. We expect such phenomena for globalcases as well. Although we can’t define the monodromy $groups_{f}$ we have some ex-amples as below. In what follows the conjectural monodromy group will be denotedby “ $G(\mathcal{F})$”.Example 1.Let $\mathcal{F}=$ { $\chi|$ primitive character mod $q,$ $\chi^{2}=1$ }. Then $L(s, f)=L(s, \chi)$ whichis the Dirichlet $L$-function, and the conductor $c_{\chi}=q$ is in the usual sense.Prediction. “ $G(\mathcal{F})"=s_{p}(\infty)$We have some evidences for this prediction. The first is the following theorem.Theorem 3.1 (Katz-Sarnak). If $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\hat{\phi})\subset(-2,2)_{f}$$W(X, \mathcal{F}, \phi)arrow\int_{-\infty}^{\infty}\phi(X)\omega(sp, x)dx$where $\omega(Sp, x)=1-\frac{\sin 2\pi x}{2\pi x}$The second evidence is Rubinstein’s experiment [R]. He investigated numericallythe distributions of $\nu_{j}(\mathcal{F}, X)(j=1,2)$ and $W(X,\mathcal{F})$ for $X\approx 10^{12}$ . He finds anexcellent fit with the $Sp(\infty)$ predictions.As the third evidence, we have the Hazelgrove phenomenon as follows. Hazel-grove numerically computed zeros in this family for moderate sized $q$ . He foundthat the zeros repel the point $s= \frac{1}{2}$ . The density of $\nu_{1}(s_{p})$ vanishes to second orderat $0$ and this is uriique to the $Sp$ symmetry. So this phenomenon is a manifestationof the symplectic symmetry.150Example 2.Let $\triangle$ be the cusp form for $SL_{2}(\mathrm{Z})$ of weight 12. Let $\mathcal{F}=$ { $\triangle\otimes\chi|\chi$ mod $q$}.We consider the family of L-functions$L(s, \triangle\otimes\chi)=\sum_{=n1}^{\infty}\frac{\tau(n)\chi(n)}{n^{\frac{11}{2}+s}}$ .We have two subfamilies $\mathcal{F}^{+}$ and $\mathcal{F}^{-}$ according to the signature of the functionalequation of $L(s, \triangle\otimes\chi)$ . Putting the conductor $c_{\triangle\otimes\chi}=q^{2}$ , the global monodromyis predicted as follows:Prediction.“$G( \mathcal{F}^{+})"=\lim_{narrow\infty}so(2n)$$‘ \zeta G(\mathcal{F}^{-})"=\lim_{narrow\infty}SO(2n-1)$The first evidence is the following theorem.Theorem 3.2 (Katz-Sarnak). If $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\hat{\phi})\subset(-1,1),$ $a\mathit{8}Xarrow\infty$$W(X, \mathcal{F}^{+}, \phi)arrow\int_{-\infty}^{\infty}\phi(X)\omega(SO(even), x)dX$ .$W(X, \mathcal{F}^{-}, \phi)arrow\int_{-\infty}^{\infty}\phi(x)\omega(so(odd), X)dX$ .We also have Rubinstein’s numerical experiments [R] with $\nu_{j}(X, \mathcal{F}^{\pm})(j=1,2)$and $W(X, \mathcal{F}^{\pm})$ with $X\approx 10^{6}$ , which agree with the $O(\infty)$ -predictions.Example 3.Let $\mathcal{F}=$ { $f|$ holo Hecke eigen cusp form for $PSL(2,$ $\mathrm{Z})\mathrm{o}\mathrm{f}$ wt $k$ }. We considerthe family of automorphic $L$-functions. We again have two subfamilies $\mathcal{F}^{+},$ $\mathcal{F}^{-}$owing to the functional equation. In fact the sign is +1 if $k\equiv 0(4)$ and $-1$ if$k\equiv 2(4)$ . By putting $c_{f}=k^{2}$ , we predict as follows.Prediction.“$G( \mathcal{F}^{+})"=\lim_{narrow\infty}so(2n)$“$G( \mathcal{F}^{-})"=\lim_{narrow\infty}SO(2n-1)$151Theorem 3.3 $(\mathrm{I}_{\mathrm{W}}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{e}\mathrm{c}-\mathrm{L}\mathrm{u}\mathrm{o}- \mathrm{S}\mathrm{a}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{k})$. If $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\hat{\phi})\subset(-1,1)_{f}a\mathit{8}Xarrow\infty$$W(X, \mathcal{F}^{+}, \phi)arrow\int_{-\infty}^{\infty}\phi(X)\omega(So(even), x)dX$$W(X, \mathcal{F}^{-}, \phi)arrow\int_{-\infty}^{\infty}\phi(x)\omega(so(odd), X)dX$If we assume the Generalized Riemann $Hypothe\mathit{8}isf$ we get it for $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\hat{\phi})\subset(-2,2)$ .Example 4.Let $\mathcal{F}=$ { $f|$ holo cusp form, wt $k$ for $\Gamma_{0}(N)$ }. Here we are interested in thesubfamily of newforms $H_{k}(N)=$ { $f|$ new form} $\subset \mathcal{F}$ . We assume that the cen-tral character of $f$ is trivial and for simplicity we also assume $N$ is prime. Let$H_{k}^{+}(N),$ $H^{-}\mathrm{A}^{r}$ (.N) be the subfamilies defined from the functional equation as thepreceding examples. Putting $c_{f}=N$ , we predict as follows.Prediction.“ $G(\mathcal{F}^{+})"=$ “ $G( \mathcal{H}_{k}^{+}(N))"=\lim_{narrow\infty}so(2n)$ .“$G(\mathcal{F}^{-})"=$ “ $G( \mathcal{H}_{k}^{-}(N))"=\lim_{narrow\infty}SO(2n-1)$ .We have the following theorem as an evidence.Theorem 3.4 $(\mathrm{I}_{\mathrm{W}}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{e}\mathrm{c}-\mathrm{L}\mathrm{u}\mathrm{o}- \mathrm{S}\mathrm{a}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{k})$. If $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\hat{\phi})\subset(-1, !)_{f}a\mathit{8}Xarrow\infty$$W(X, H_{k}^{+}(N),$ $\phi)arrow\int_{-\infty}^{\infty}\phi(x)\omega(SO(even), x)dx$$W(X, H_{k^{-}}(N),$ $\phi)arrow\int_{-\infty}^{\infty}\phi(X)\omega(so(odd), X)dX$If we assume the Generalized Riemann Hypothesis, we get it for $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\hat{\phi})\subset$$(-2,2)$ .Example 5.Let $\mathcal{F}=$ { $\vee^{2}f|f\in$ Example 3}. We consider the symmetric squared L-functions $L(s, \vee^{2}f)$ . It is equal to $L(s,\overline{f})$ for $\overline{f}$ a self-dual cusp form on $GL_{3}$ . Ithas an Euler product of degree 3. By putting $c_{\vee^{2}f}=k^{2}$ , we havePrediction. “ $G(\mathcal{F})"=Sp(\infty)$The following theorem is proved as an evidence.152Theorem 3.5 (Iwaniec-Luo-Sarnak $\rangle$ . If $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\hat{\phi})\subset(-1,1)$, as $Xarrow\infty$$W(X, \mathcal{F}, \phi)arrow\int_{-\infty}^{\infty}\phi(X)\omega(Sp, X)dx$If we $a\mathit{8}sume$ the Generalized Riemann $Hypothe\mathit{8}i\mathit{8}_{f}$ we get it for $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\hat{\phi})\subset(-\frac{4}{3}, \frac{4}{3})$ .From the examples given above, we have the following general conjecture.Density Conjecture.$W(X, \mathcal{F}, \phi)$ converges to the claimed density without any restriction on $\hat{\phi}$ .4. APPLICATIONSIn the settings given in Example 4, we have some remarkable applications. TheDensity Conjecture implies that$\frac{\#\{f\in \mathcal{F}|c_{f}\leq x,\epsilon f=1,L(\frac{1}{2},f)\neq 0\}}{\#\{f\in \mathcal{F}|c_{f}\leq x,\epsilon f=1\}}arrow 1$as $Xarrow\infty$ . Towards the Density Conjecture, we have some partial results asfollows:Theorem 4.1 $(\mathrm{I}_{\mathrm{W}}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{e}\mathbb{C}-\mathrm{S}\mathrm{a}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{k}- \mathrm{L}\mathrm{u}\mathrm{o})$. Assume the Generalized Riemann Hy-$potheSi\mathit{8}_{f}$ then$\frac{\#\{f\in H_{2}^{+}(N)|L(\frac{1}{2},f)\neq 0\}}{\#\{f\in H_{2}^{+}(N)\}}>\frac{9}{16}$Theorem 4.2 (Iwaniec-Sarnak).$\lim_{Narrow\infty}\frac{\#\{f\in H_{2}^{+}(N)|L(\frac{1}{2},f)\geq\frac{1}{(\log N)^{2}}\}}{\#\{f\in H_{2}^{+}(N)\}}\geq\frac{1}{2}$Theorem 4.3 (Iwaniec-Sarnak).If Theorem 4.2 holds with any $C> \frac{1}{2}$ in place of $\frac{1}{2}\rangle$ there are no Siegel zeros!REFERENCES[ILS] H. Iwaniec, W. Luo and P. Sarnak, Low lying zeros of $L$ -functions of modular forms(preprint).[IS] H. Iwaniec and P. Sarnak, The non-vanishing of central values of automorphic L-functionsand Siegel zeros (preprint).[KS1] N. Katz and P. Sarnak, Random matrices, Frobenius eigenvalues and monodromy (1998),AMS (to appear).[KS2] N. Katz and P. Sarnak, Zeros of zeta $functions_{\mathit{1}}$ their spacings and their spectral nature(preprint).[O] A. M. Odlyzko,, On the $\dot{d}$istribution of spacings between zeros of the zeta function., Math.of Computation 48 (1987), 273-308.[R] M. Rubinstein, Evidence for a spectral interpretation of the zeros of L-functions.[RS] Z. Rudnick and P. Sarnak, The $n$-level correlations of zeros of the zeta function,, C. R.Acad. Sci. Paris 319 (1994), 1027-1032.DEPT OF MATH., KEIO UNIVERSITY, 223-8522 JAPAN153

ON THE RESULTS OF SARNAK-RUDNICK-KATZ-IWANIEC-LUO ON ZEROS OF ZETA FUNCTIONS (Number Theory and its Applications)

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ON THE RESULTS OF SARNAK-RUDNICK-KATZ-IWANIEC-LUO ON ZEROS OF ZETA FUNCTIONS (Number Theory and its Applications)

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