Shifted convolution sums play a prominent role in analytic number theory. We investigate pointwise bounds, mean-square bounds, and average bounds for shifted convolution sums for Hecke eigenforms.
The Zagier $L$-series encode data of real quadratic fields. We study the average size of these $L$-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.
This is an expanded writeup of a talk given by the second author at Erik Balslevs 75th birthday conference on October 1-2, 2010 at Aarhus University. We summarize our work on Fermis golden rule and higher order phenomena for hyperbolic manifolds. A t
opic which occupied the last part of Erik Balslevs research.
Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve $L$-functions. Two of these conjectures relate to the asymptoti
c growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture in a refined version that also allows restrictions on the location of the cusps.
For $Gamma={hbox{PSL}_2( {mathbb Z})}$ the hyperbolic circle problem aims to estimate the number of elements of the orbit $Gamma z$ inside the hyperbolic disc centered at $z$ with radius $cosh^{-1}(X/2)$. We show that, by averaging over Heegner point
s $z$ of discriminant $D$, Selbergs error term estimate can be improved, if $D$ is large enough. The proof uses bounds on spectral exponential sums, and results towards the sup-norm conjecture of eigenfunctions, and the Lindelof conjecture for twists of the $L$-functions attached to Maa{ss} cusp forms.
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 exp
licit expression. Moreover, we prove that $e_alpha(s)$ has a limiting distribution.
It is well known that the angles in a lattice acting on hyperbolic $n$-space become equidistributed. In this paper we determine a formula for the pair correlation density for angles in such hyperbolic lattices. Using this formula we determine, among
other things, the asymptotic behavior of the density function in both the small and large variable limits. This extends earlier results by Boca, Pasol, Popa and Zaharescu and Kelmer and Kontorovich in dimension 2 to general dimension $n$. Our proofs use the decay of matrix coefficients together with a number of careful estimates, and lead to effective results with explicit rates.
The hyperbolic lattice point problem asks to estimate the size of the orbit $Gamma z$ inside a hyperbolic disk of radius $cosh^{-1}(X/2)$ for $Gamma$ a discrete subgroup of $hbox{PSL}_2(R)$. Selberg proved the estimate $O(X^{2/3})$ for the error term
for cofinite or cocompact groups. This has not been improved for any group and any center. In this paper local averaging over the center is investigated for $hbox{PSL}_2(Z)$. The result is that the error term can be improved to $O(X^{7/12+epsilon})$. The proof uses surprisingly strong input e.g. results on the quantum ergodicity of Maa{ss} cusp forms and estimates on spectral exponential sums. We also prove omega results for this averaging, consistent with the conjectural best error bound $O(X^{1/2+epsilon})$. In the appendix the relevant exponential sum over the spectral parameters is investigated.
The problem of quantum unique ergodicity (QUE) of weight 1/2 Eisenstein series for {Gamma}_0(4) leads to the study of certain double Dirichlet series involving GL2 automorphic forms and Dirichlet characters. We study the analytic properties of this f
amily of double Dirichlet series (analytic continuation, convexity estimate) and prove that a subconvex estimate implies the QUE result.
We identify the quantum limits of scattering states for the modular surface. This is obtained through the study of quantum measures of non-holomorphic Eisenstein series away from the critical line. We provide a range of stability for the quantum unique ergodicity theorem of Luo and Sarnak.
