Murmurations and explicit formulas

We provide a theoretical explanation for an observation of S. J. Miller that if L(s,E) is an elliptic curve L-function for which L(1/2, E) is nonzero, then the lowest lying zero of L(s,E) exhibits a repulsion from the critical point which is not explained by the standard Katz-Sarnak heuristics. We establish a similar result in the case of first-order vanishing.

In this paper we determine the limiting distribution of the image of the Eichler--Shimura map or equivalently the limiting joint distribution of the coefficients of the period polynomials associated to a fixed cusp form. The limiting distribution is shown to be the distribution of a certain transformation of two independent random variables both of which are equidistributed on the circle $\mathbb{R}/\mathbb{Z}$, where the transformation is connected to the additive twist of t...

It is believed that, in the limit as the conductor tends to infinity, correlations between the zeros of elliptic curve $L$-functions averaged within families follow the distribution laws of the eigenvalues of random matrices drawn from the orthogonal group. For test functions with restricted support, this is known to be the true for the one- and two-level densities of zeros within the families studied to date. However, for finite conductor Miller's experimental data reveal an...

In this series, we investigate the calculation of mean values of derivatives of Dirichlet $L$-functions in function fields using the analogue of the approximate functional equation and the Riemann Hypothesis for curves over finite fields. The present paper generalizes the results obtained in the first paper. For $\mu\geq1$ an integer, we compute the mean value of the $\mu$-th derivative of quadratic Dirichlet $L$-functions over the rational function field. We obtain the full ...

As we have shown several years ago [Y2], zeros of $L(s, \Delta )$ and $L^(2)(s, \Delta )$ can be calculated quite efficiently by a certain experimental method. Here $\Delta$ denotes the cusp form of weight 12 with respect to SL$(2, Z)$ and $L(s, \Delta )$ (resp. $L^(2)(s, \Delta )$) denotes the standard (resp. symmetric square) $L$-function attached to $\Delta$. The purpose of this paper is to show that this method can be applied to a wide class of $L$-functions so that we ca...

We study the low-lying zeros of various interesting families of elliptic curve L-functions. One application is an upper bound on the average analytic rank of the family of all elliptic curves. The upper bound obtained is less than two, which implies that a positive proportion of elliptic curves over the rationals have algebraic rank equal to analytic rank and finite Tate-Shafarevich group. These results are conditional on the Generalized Riemann Hypothesis.

We consider heuristic predictions for small non-zero algebraic central values of twists of the $L$-function of an elliptic curve $E/\mathbb{Q}$ by Dirichlet characters. We provide computational evidence for these predictions and consequences of them for instances of an analogue of the Brauer-Siegel theorem.

Let $f$ be a normalized Hecke-Maass cusp form of weight zero for the group $SL_2(\mathbb Z)$. This article presents several quantitative results about the distribution of Hecke eigenvalues of $f$. Applications to the $\Omega_{\pm}$-results for the Hecke eigenvalues of $f$ and its symmetric square sym$^2(f)$ are also given.

In this paper, we consider the mean value of the product of two real valued multiplicative functions with shifted arguments. The functions $F$ and $G$ under consideration are close to two nicely behaved functions $A$ and $B$, such that the average value of $A(n-h)B(n)$ over any arithmetic progression is only dependent on the common difference of the progression. We use this method on the problem of finding mean value of $K(N)$, where $K(N)/\log N$ is the expected number of pr...

In this paper an explicit formula is given for a sequence of numbers. The positivity of this sequence of numbers implies that zeros in the critical strip of the Euler product of Hecke polynomials, which are associated with the space of cusp forms of weight $k$ for Hecke congruence subgroups, lie on the critical line.