On the mean value of a kind of Zeta functions

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.