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Abstract

Let $d_{\alpha ,\beta }(n)=\sum _{ n=kl,\,\alpha l< k\leq \beta l}1$ be the number of ways of factoring $n$ into two almost equal integers. For fixed rational numbers $\alpha >0$ and $\beta >0$, we consider a zeta function of the type $\zeta _{\alpha ,\beta }(s)=\sum _{n=1}^{\infty }{d_{\alpha ,\beta }(n)}/{n^{s}}$ 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 a result on the distribution of primitive Pythagorean triangles.