On the mean value of a kind of Zeta functions

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$\Re s>1.$Ithasananalyticcontinuationto$\Re s>1/3.$Wegetanasymptoticformulaforthemeansquareof$\zeta_{\alpha,\beta}(s)$inthestrip$1/2<\Re s<1$. As an application, we improve an result on the distribution of primitive Pythagorean triangles.