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Register a new userOn the variance of the error term in the hyperbolic circle problem
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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.
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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 $ba=(a_1,a_2,ldots,a_k)$, where $a_j (j=1,ldots,k)$ are positive integers such that $a_1 leq a_2 leq cdots leq a_k$. Let $d(ba;n)=sum_{n_1^{a_1}cdots n_k^{a_k}=n}1$ and $Delta(ba;x)$ be the error term of the summatory function of $d(ba;n)$. In this paper we show an asymptotic formula of the mean square of $Delta(ba;x)$ under a certain condition. Furthermore, in the cases $k=2$ and 3, we give unconditional asymptotic formulas for these mean squares.
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 $sqrt{n}$ and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including $sum P_k(n)^2 e^{-n/X}$ and the Laplace transform $int_0^infty P_k(t)^2 e^{-t/X}dt$, in dimensions $k geq 3$. We also obtain main terms and power-saving error terms for the sharp sums $sum_{n leq X} P_k(n)^2$, along with similar results for the sharp integral $int_0^X P_3(t)^2 dt$. This includes producing the first power-saving error term in mean square for the dimension-three Gauss circle problem.
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 series has meromorphic continuation to $mathbb{C}$. Using this series, we prove that the Laplace transform of $P_2(n)^2$ satisfies $int_0^infty P_2(t)^2 e^{-t/X} , dt = C X^{3/2} -X + O(X^{1/2+epsilon})$, which gives a power-savings improvement to a previous result of Ivic [Ivic1996]. Similarly, we study the meromorphic continuation of the Dirichlet series associated to the correlations $r_2(n+h)r_2(n)$, where $h$ is fixed and $r_2(n)$ denotes the number of representations of $n$ as a sum of two squares. We use this Dirichlet series to prove asymptotics for $sum_{n geq 1} r_2(n+h)r_2(n) e^{-n/X}$, and to provide an additional evaluation of the leading coefficient in the asymptotic for $sum_{n leq X} r_2(n+h)r_2(n)$.
In 1956, Tong established an asymptotic formula for the mean square of the error term in the summatory function of the Piltz divisor function $d_3(n).$ The aim of this paper is to generalize Tongs method to a class of Dirichlet series that satisfy a functional equation. As an application, we can establish the asymptotic formulas for the mean square of the error terms for a class of functions in the well-known Selberg class. The Tong-type identity and formula established in this paper can be viewed as an analogue of the well-known Voronois formula.
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