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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 points $z$ 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
Given a newform f, we extend Howards results on the variation of Heegner points in the Hida family of f to a general quaternionic setting. More precisely, we build big Heegner points and big Heegner classes in terms of compatible families of Heegner
The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to $P_k(n)^2$, where $P_k(n)$ is the discrepancy between the volume of the $k$-dimensional sphere of radius $sqrt{n
The Gauss circle problem concerns the difference $P_2(n)$ between the area of a circle of radius $sqrt{n}$ and the number of lattice points it contains. In this paper, we study the Dirichlet series with coefficients $P_2(n)^2$, and prove that this se
If the $ell$-adic cohomology of a projective smooth variety, defined over a $frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $ge 1$, then any model over the ring of integers of $K$ has a $k$-rational point. This sli