Data Associated to the Pair Correlation
We present the numerics for three \(L\)-functions. We include various visualizations from the data generated by the zeros of \(\zeta\) and two \(\mathrm{GL}_2\) \(L\)-functions. The first \(\mathrm{GL}_2\) \(L\)-function we consider is \(L(s,\Delta)\), the \(L\)-function associated with the modular discriminant \begin{align*} \Delta(z) = e^{2\pi i z}\prod_{n=1}^{\infty}(1-e^{2\pi i n z})^{24} = \sum_{n=1}^{\infty}\tau(n)e^{2\pi i nz}, \end{align*} which is a newform of weight 12 and level 1. The second one is \(L(s,E),\) where \(E/\mathbb{Q}\) is the elliptic curve over \(\mathbb{Q}\) defined by \begin{align*} E: y^2 +y = x^3-x. \end{align*} These \(L\)-functions were studied computationally by Katz and Sarnak in [3]. For more information on the Elliptic curve, see https://www.lmfdb.org/EllipticCurve/Q/11/a/3. Additionally, these \(L\)-function's zeros are symmetric over the real line. To conduct our numerical experiments, we obtain the first million non-trivial zeros with positive ordinates of these \(L\)-functions. For \(\zeta(s)\), we use the zeros computed by Platt [6] available through Zeros of \(\zeta(s)\) [2]. For the other \(L\)-functions, we use Rubinstein's lcalc package [7] to compute their zeros. Although we do not use all the zeros computed by Platt, we remark that our data strongly reflects our conjectures. For each \(L\)-function, we compute the following sets \begin{equation}\label{def:S(pi,X,T)} \mathcal{S}(\pi,X,T) := \{\Re\mathcal{S}_{\pi}(X,T,\rho_{\pi}'): -T \leqslant \Im \rho \leqslant T\} \end{equation} for many \(X.\) Recall, we define \begin{align*} \mathcal{S}_{\pi}(X,T,\rho_{\pi}') := \sum_{|\text{Im} \rho_{\pi}| \leq T} X^{\rho_{\pi}-\rho_{\pi'}}w(\rho_{\pi}-\rho_{\pi}') \end{align*} We denote the standard deviation of this set by \begin{equation}\label{def:SD} \mathcal{S}_{\text{SD},\pi}(X,T):= \bigg(\frac{1}{N(T)}\sum_{|\text{Im} \rho_{\pi}|\leqslant T} \Big(\Re\mathcal{S}_{\pi}(X,T,\rho'_{\pi}) -\mathcal{S}_{\text{av},\pi}(X,T)\Big)^2\bigg)^{\frac{1}{2}}. \end{equation} In general, we choose \(T\) be the ordinate of the millionth zero of the \(L\)-function: \(T \approx 600269\) for \(\zeta\), \(T \approx 319387\) for \(L(s,\Delta)\), and \(T \approx 273499\) for \(L(s,E)\). The relevant statistics and graphs are then created for these sets. Explicitly, we compute the mean, the standard deviation, various moments, a discrete approximation of the probability density function, and a discrete approximation of the PCS from these sets. In this website, we slightly change our notation from the paper [1]. When \(\pi\) is the trivial representation corresponding to \(\zeta\), we define for \(\rho'\) with \(\zeta(\rho') =0\) \begin{align*}\label{S_zeta} \mathcal{S}(X,T,\rho') := \mathcal{S}_{\pi}(X,T,\rho'). \end{align*} For the other \(L\)-functions, we replace the \(\pi\) in \(S_{\pi}(X,T,\rho')\) for the arithmetic object they are associated with, specifically, \(\Delta\) and \(E.\) We follow the same changes for \(\mathcal{S}_{\text{av},\pi}(X,T), \mathcal{S}_{\text{SD},\pi}(X,T),\) and \(\mathcal{S}(\pi,X,T).\)Data Associated to Triple Correlation
In this project, we are also interested in distribution of non-trivial zeros of automorphic \(L\)-functions with respect to their triple correlation. Let \(\Psi \in C_{c}^{\infty}(0,\infty)\) be a fixed non-negative compactly supported smooth function. Let \(X_1,X_2,T \geqslant 2\) and consider the sums \begin{align} \widehat{\mathcal{S}}_{\pi,\Psi}(X_1,X_2,T,\rho_{\pi}) &:= \sum_{\substack{ \lvert \Im{\rho_{\pi,1}} \rvert \leqslant T}} \sum_{\substack{ \lvert \Im{\rho_{\pi,2}} \rvert \leqslant T}}\hat{\Psi}_{X_1}(\rho_{\pi,1} - \rho_{\pi}) \hat{\Psi}_{X_2}(\rho_{\pi} - \rho_{\pi,2}), \label{eqn:S triple}\tag{1} \end{align} where \(\rho_{\pi,1}, \rho_{\pi,2}\) vary over the non-trivial zeros of \(L(s, \pi)\). Set \(X_1 = T^{m\alpha_1}\) and \(X_2 = T^{m\alpha_2}\) for \(\alpha_1 > 0\) and \(\alpha_2>0\). We find that the moments of the random variable defined behave differently when \(\alpha_1 \neq \alpha_2\) and when \(\alpha_1 = \alpha_2\). With certain restrictions on \(\alpha_1\) and \(\alpha_2\), we find that when \(\alpha_1 = \alpha_2\) our random variables have the same moments as Chi-square distribution of two degrees of freedom; on the other hand, when \(\alpha_1 \neq \alpha_2\) they have the same moments as Laplacian with mean zero and scaling factor one. See [1, Theorem 1.16] for a complete statement and more infromation. Due to that both of these distribution have heavy tails, we unable to conclude the distribution from these moments; however, with preliminary computations we find that the distributions do seem to act as expected. For these preliminary computations, we compute the following sum \begin{equation*} \mathcal{S}_{\pi,\Psi}(\rho_{\pi}) =\sum_{n_1,n_2 =1}^{\infty} \frac{\Psi_{X_1}(n_1) \Psi_{X_2}(n_2) \Lambda_{\pi}(n_1) \Lambda_{\widetilde{\pi}}(n_2)}{(n_1n_2)^{\frac{1}{2}}} \bigg( \frac{n_2}{n_1} \bigg)^{i \gamma_{\pi}}. \end{equation*} This sum is related to our orginal double sum by an explicit formula. See [1,Lemma 8.3] for more details. In our computations, we take \(\Psi\) to be a smooth approximation of the indicator function of the interval \((\frac{1}{2}, \frac{3}{4})\). We further take \(L(s,\pi)\) to be \(\zeta(s)\) and use the first 30 million zeros of \(\zeta(s)\). We include some visualizations of our data under the Triple Correlation tab. We are currently working on our data set made up the sums defined by \(\textbf{(1)}\).