Some integrals of hypergeometric functions

We consider a certain definite integral involving the product of two classical hypergeometric functions having complicated arguments. We show the surprising fact that this integral does not depend on the parameters of the hypergeometric functions. © 2017, Akadémiai Kiadó, Budapest, Hungary

Triple product integrals and Rankin-Selberg L-functions

We show that a certain spectral sum of the product of two triple product integrals of automorphic forms (where the first factor involves the classical theta-function, a fixed Maass cusp form $u_1$ of weight 0 and a varying Maass cusp form of weight 1/2 , and the second factor is similar, but contains another fixed Maass cusp form $u_2$ of weight 0 in place of $u_1$) can be expressed as an integral involving the Rankin-Selberg L-function of $u_1$ and $u_2$. In the special case $u_1$ = $u_2$ our formula reduces to a spectral reciprocity identity which (in slightly different form) was proved recently by Humphries-Khan ([H-K]) and Kwan ([Kw]).Comment: 44 page

Entropic Distance for Nonlinear Master Equation

More and more works deal with statistical systems far from equilibrium, dominated by unidirectional stochastic processes augmented by rare resets. We analyze the construction of the entropic distance measure appropriate for such dynamics. We demonstrate that a power-like nonlinearity in the state probability in the master equation naturally leads to the Tsallis (Havrda-Charv\'at, Acz\'el-Dar\'oczy) q-entropy formula in the context of seeking for the maximal entropy state at stationarity. A few possible applications of a certain simple and linear master equation to phenomena studied in statistical physics are listed at the end.Comment: Talk given by T.S.Bir\'o at BGL 2017, Gy\"ongy\"os, Hungar

On the geometric trace of a generalized Selberg trace formula

A certain generalization of the Selberg trace formula was proved by the first named author in 1999. In this generalization instead of considering the integral of $K(z,z)$ (where $K(z,w)$ is an automorphic kernel function) over the fundamental domain, one considers the integral of $K(z,z)u(z)$, where $u(z)$ is a fixed automorphic eigenfunction of the Laplace operator. This formula was proved for discrete subgroups of $PSL(2,\mathbb{R})$, and just as in the case of the classical Selberg trace formula it was obtained by evaluating in two different ways ("geometrically" and "spectrally") the integral of $K(z,z)u(z)$. In the present paper we work out the geometric side of a further generalization of this generalized trace formula: we consider the case of discrete subgroups of $PSL(2,\mathbb{R})^n$ where $n>1$. Many new difficulties arise in the case of these groups due to the fact that the classification of conjugacy classes is much more complicated for $n>1$ than in the case $n=1$.Comment: 38 pages, 0 figure

Strong Characterizing Sequences in Simultaneous Diophantine Approximation

Answering a question of Liardet. we prove that if 1, alpha(1), alpha(2)...... alpha(t) are real numbers linearly independent over the rationals, then there is an infinite subset A of the positive integers such that for real beta, we have (|| || denotes the distance to the nearest integer) Sigma(nequivalent toA)||nbeta||<infinity if and only if beta is a linear combination with integer coefficients of 1, alpha(1), alpha(2,)..., alpha(t). The proof combines elementary ideas with a deep theorem of Freiman on set addition. Using Freiman's theorem, we prove a lemma on the structure of Bohr sets, which may have independent interest. (C) 2002 Elsevier Science (USA). All rights reserved

On Polynomials over Prime Fields Taking Only Two Values on the Multiplicative Group

AbstractLet p>2 be a prime, denote by Fp the field with ∣Fp∣=p, and let F*p=Fp\{0}. We prove that if fϵFp[x] and f takes only two values on F*p, then (excluding some exceptional cases) the degree of f is at least 34(p−1)

LOCAL AVERAGE OF THE HYPERBOLIC CIRCLE PROBLEM FOR FUCHSIAN GROUPS

Let $\Gamma\subseteq PSL(2,{\bf R})$ be a finite volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $\Gamma$-orbit of $z$ in a hyperbolic circle around $w$ of radius $R$, where $z$ and $w$ are given points of the upper half plane and $R$ is a large number. An estimate with error term $e^{{2\over 3}R}$ is known, and this has not been improved for any group. Recently Risager and Petridis proved that in the special case $\Gamma =PSL(2,{\bf Z})$ taking $z=w$ and averaging over $z$ in a certain way the error term can be improved to $e^{\left({7\over {12}}+\epsilon\right)R}$. Here we show such an improvement for a general $\Gamma$, our error term is $e^{\left({5\over 8}+\epsilon\right)R}$ (which is better that $e^{{2\over 3}R}$ but weaker than the estimate of Risager and Petridis in the case $\Gamma =PSL(2,{\bf Z})$). Our main tool is our generalization of the Selberg trace formula proved earlier.Comment: Accepted by Mathematik