On the mean value of a kind of Zeta functions

Abstract

Let $d_{\alpha ,\beta }(n)=\sum _{\begin{array}{c}n=kl\alpha l<k\le \beta l\end{array}}1$ 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 ${\zeta }_{\alpha ,\beta }(s)=\sum _{n=1}^{\infty }\frac{d_{\alpha ,\beta }(n)}{n^s}$ 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.