On the mean value of a kind of Zeta functions

TL;DR

This paper studies a special Zeta function related to factorizations into nearly equal parts, providing asymptotic formulas for its mean square and applying results to the distribution of primitive Pythagorean triangles.

Contribution

It introduces a new Zeta function based on almost equal factorizations and derives its mean square asymptotics, extending analytic continuation and applying to Pythagorean triangle distribution.

Findings

Derived asymptotic formula for the mean square of the Zeta function

Extended the analytic continuation of the Zeta function to Re s > 1/3

Improved understanding of primitive Pythagorean triangle distribution

Abstract

Let $d_{\alpha, \beta}(n)=\sum\limits_{\substack{n=kl \alpha l<k\leq\beta l}}1$ be the number of ways of factoring n into two almost equal integers. For rational numbers $0<\alpha <\beta $, we consider the following Zeta function $\zeta_{\alpha,\beta}(s)=\sum\limits_{n=1}^{\infty}\frac{d_{\alpha, \beta}(n)}{n^{s}}$ for $\Re s>1.$ It has an analytic continuation to $\Re s>1/3.$ We get an asymptotic formula for the mean square of $\zeta_{\alpha,\beta}(s)$ in the strip $1/2<\Re s<1$. As an application, we improve an result on the distribution of primitive Pythagorean triangles.