A segment of Euler product associated to a certain Dirichlet series

Abstract

In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function $\ddot{I}{ƒ}_{\hat{I}}\pm (n):={\sum }_{d|n}{d}^{\hat{I}}\pm$. We obtain an exact identity relating the Dirichlet series $\hat{I}\P (s)\hat{I}\P (s-\hat{I}\pm )$ and a segment of the Euler product attached to it. Specifically, our main theorems are valid in the critical strip.