A segment of Euler product associated to a certain Dirichlet series

In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function ${\sigma }_{\alpha }(n):={\sum }_{d\mathrm{\mid }n}{d}^{\alpha }$. 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.