On the variance of the nodal volume of arithmetic random waves

Rudnick and Wigman (Ann. Henri Poincar{e}, 2008; arXiv:math-ph/0702081) conjectured that the variance of the volume of the nodal set of arithmetic random waves on the $d$-dimensional torus is $O(E/mathcal{N})$, as $Etoinfty$, where $E$ is the energy and $mathcal{N}$ is the dimension of the eigenspace corresponding to $E$. Previous results have established this with stronger asymptotics when $d=2$ and $d=3$. In this brief note we prove an upper bound of the form $O(E/mathcal{N}^{1+alpha(d)-epsilon})$, for any $epsilon>0$ and $dgeq 4$, where $alpha(d)$ is positive and tends to zero with $d$. The power saving is the best possible with the current method (up to $epsilon$) when $dgeq 5$ due to the proof of the $ell^{2}$-decoupling conjecture by Bourgain and Demeter.

A spectral universality theorem for Maass $L$-functions

We show that for a positive proportion of Laplace eigenvalues $lambda_j$ the associated Hecke-Maass $L$-functions $L(s,u_j)$ approximate with arbitrary precision any target function $f(s)$ on a closed disc with center in $3/4$ and radius $r<1/4$. The main ingredients in the proof are the spectral large sieve of Deshouillers-Iwaniec and Sarnaks equidistribution theorem for Hecke eigenvalues.

Almost periodic functions and hyperbolic counting

In this paper we prove the existence of asymptotic moments, and an estimate on the tails of the limiting distribution, for a specific class of almost periodic functions. Then we introduce the hyperbolic circle problem, proving an estimate on the asymptotic variance of the remainder that improves a result of Chamizo. Applying the results of the first part we prove the existence of limiting distribution and asymptotic moments for three functions that are integrat

On the variance of the error term in the hyperbolic circle problem

Let $e(s)$ be the error term of the hyperbolic circle problem, and denote by $e_alpha(s)$ the fractional integral to order $alpha$ of $e(s)$. We prove that for any small $alpha>0$ the asymptotic variance of $e_alpha(s)$ is finite, and given by an explicit expression. Moreover, we prove that $e_alpha(s)$ has a limiting distribution.