Some Applications of Laplace Transforms in Analytic Number Theory

TL;DR

This paper explores how Laplace transforms can be applied to various problems in analytic number theory, including classical problems, moments of the zeta function, and functional equations, providing new insights and methods.

Contribution

It introduces novel applications of Laplace transforms to classical and modern problems in analytic number theory, expanding the toolkit for researchers.

Findings

New applications of Laplace transforms to the circle and divisor problems

Analysis of moments of the Riemann zeta function using Laplace transforms

Discussion of functional equations related to Stanković's work

Abstract

In this overview paper, presented at the meeting DANS14, Novi Sad, July3-7, 2014, we give some applications of Laplace transforms to analytic number theory. These include the classical circle and divisor problem, moments of $|\zeta(1/2+it)|$, and a discussion of two functional equations connected to a work of Prof. Bogoljub Stankovi\'c.