Related papers: Symmetry and short interval mean-squares

We study the symmetry in short intervals of arithmetic functions with non-negative exponential sums.

We analyze the situation which is related to zonal spherical functions of type $A_n$ and obtain a generalization of Selberg integral.

Let $\varrho$ be a complex number and let $f$ be a multiplicative arithmetic function whose Dirichlet series takes the form $\zeta(s)^\varrho G(s)$, where $G$ is associated to a multiplicative function $g$. The classical Selberg-Delange…

Let $1/2\leq\beta<1$, $p$ be a generic prime number and $f_\beta$ be a random multiplicative function supported on the squarefree integers such that $(f_\beta(p))_{p}$ is an i.i.d. sequence of random variables with distribution…

We obtain an explicit formula for a certain weighted enumeration of lozenge tilings of a hexagon with an arbitrary triangular hole. The complexity of our expression depends on the distance from the hole to the center of the hexagon. This…

An arithmetic function $f$ is called a {\it sieve function of range} $Q$, if its Eratosthenes transform $g=f\ast\mu$ is supported in $[1,Q]\cap\N$, where $g(q)\ll_{\varepsilon} q^{\varepsilon}$ ($\forall\varepsilon>0$). Here, we study the…

Quantitative formulations of Fefferman's counterexample for the ball multiplier are naturally linked to square function and vector-valued estimates for directional singular integrals. The latter are usually referred to as Meyer-type lemmas…

We propose a definition of the weighted $\sigma_k$-curvature of a smooth metric measure space and justify it in two ways. First, we show that the weighted $\sigma_k$-curvature prescription problem is governed by a fully nonlinear second…

In this article we consider the elliptic Selberg integral, which is a BC_n symmetric multivariate extension of the elliptic beta integral. We categorize the limits that are obtained as p->0, for given behavior of the parameters as p->0.…

We prove an $\mathbb F_p$-Selberg integral formula, in which the $\mathbb F_p$-Selberg integral is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by analogy between…

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and let $E(T)$ denote the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) := E(t) - 2\pi\Delta^*(t/(2\pi))$ with $\Delta^*(x)…

An overview of the classical Rankin-Selberg problem involving the asymptotic formula for sums of coefficients of holomorphic cusp forms is given. We also study the function $\Delta(x;\xi) (0\le\xi\le1)$, the error term in the Rankin-Selberg…

We study the convolution function $$ C[f(x)] := \int_1^x f(y)f({x\over y}) {{\rm d} y\over y} $$ when $f(x)$ is a suitable number-theoretic error term. Asymptotics and upper bounds for $C[f(x)]$ are derived from mean square bounds for…

A classical problem in number theory is showing that the mean value of an arithmetic function is asymptotic to its mean value over a short interval or over an arithmetic progression, with the interval as short as possible or the modulus as…

We define a weighted multiplicity function for closed geodesics of given length on a finite area Riemann surface. These weighted multiplicities appear naturally in the Selberg trace formula, and in particular their mean square plays an…

The ostrowski inequality expresses bounds on the deviation of a function from its integral mean. The aim of this paper is to establish a new inequality using weight function which generalizes the inequalities of Dragomir, Wang and Cerone…

We introduce a generalization of the Stirling numbers via symmetric functions involving two weight functions. The resulting extension unifies previously known Stirling-type sequences with known symmetric function forms, as well as other…

In this paper we obtain a weighted average formula for special values of $L$-functions attached to normalized elliptic modular forms of weight $k$ and full level. These results are obtained by studying the pullback of a Siegel Eisenstein…

The standard twist $F(s,\alpha)$ of $L$-functions $F(s)$ in the Selberg class has several interesting properties and plays a central role in the Selberg class theory. It is therefore natural to study its finer analytic properties, for…

Under the generalized Riemann hypothesis, we use Beurling-Selberg extremal functions to bound the mean and mean square of the argument of Dirichlet $L$-functions to a large prime modulus $q$. As applications, we give alternative proofs of…