A segment of Euler product associated to a certain Dirichlet series

TL;DR

This paper investigates a specific segment of the Euler product linked to a Dirichlet series involving the generalized divisor function, providing exact identities within the critical strip.

Contribution

It establishes an exact identity connecting the Dirichlet series of the generalized divisor function with a segment of the Euler product, extending previous work to the critical strip.

Findings

Derived an exact identity relating the Dirichlet series and Euler product segment.

Extended the analysis of divisor functions to the critical strip.

Provided new insights into the structure of associated Euler products.

Abstract

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