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Register a new userThe Laplace Transform of the Second Moment in the Gauss Circle Problem
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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 $sum_{n leq X} r_2(n+h)r_2(n)$.
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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.
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.
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.
We prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment of central values of L-functions of any two (possibly equal) fixed cusp forms f, g twisted by all primitive characters modulo q, valid for all sufficiently factorable q including 99.9% of all admissible moduli. The two key ingredients are a careful spectral analysis of a potentially highly unbalanced shifted convolution problem in Hecke eigenvalues and power-saving bounds for sums of products of Kloosterman sums where the length of the sum is below the square-root threshold of the modulus. Applications are given to simultaneous non-vanishing and lower bounds on higher moments of twisted L-functions.
For a fairly general family of L-functions, we survey the known consequences of the existence of asymptotic formulas with power-sawing error term for the (twisted) first and second moments of the central values in the family. We then consider in detail the important special case of the family of twists of a fixed cusp form by primitive Dirichlet characters modulo a prime q, and prove that it satisfies such formulas. We derive arithmetic consequences: - a positive proportion of central values L(f x chi, 1/2) are non-zero, and indeed bounded from below; - there exist many characters chi for which the central L-value is very large; - the probability of a large analytic rank decays exponentially fast. We finally show how the second moment estimate establishes a special case of a conjecture of Mazur and Rubin concerning the distribution of modular symbols.
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