On the mean value of a kind of zeta functions

Kui Liu¹

¹ Department of Mathematics Qingdao University 266071, Qingdao, Shandong, China

Acta Arithmetica, Tome 166 (2014) no. 1, pp. 33-54

Résumé

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 $1/2\Re s1$. As an application, we improve a result on the distribution of primitive Pythagorean triangles.

Détail

DOI : 10.4064/aa166-1-4

Keywords: alpha beta sum alpha leq beta number ways factoring almost equal integers fixed rational numbers alpha beta consider zeta function type zeta alpha beta sum infty alpha beta has analytic continuation get asymptotic formula mean square zeta alpha beta strip application improve result distribution primitive pythagorean triangles