On the Explicit Expression of an Extended Version of Riemann Zeta Function

Abstract

In this paper, we focus on the explicit expression of an extended version of Riemann zeta function. We use two different methods, Mellin inversion formula and Cauchy's residue theorem, to calculate a Mellin-Barnes type integral of the analytic function regarding $z$: $\Gamma(z)\Gamma(s-z)u^{-z}$ ($u\in (0,1)$, $s\in \mathbb{C}$). We provide the necessary background on the analytic properties of Gamma and Riemann zeta function to confirm the absolute convergence of this Mellin-Barnes integral. Next, we represent the extended version of Riemann zeta function $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}{(m+n)^{-s}}$ using the following complex integral where the real part of $s$ is larger than 2 and $c>1$ is chosen to make $\Re(s)-c$ larger than 1. $$\Gamma(s)\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}{(m+n)^{-s}}=\frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} \zeta(z) \zeta(s - z) \Gamma(z) \Gamma(s - z) \, dz$$ We provide the evaluation of this integral by changing the integration path from straight line $\Re(z)=c$ into a rectangular contour whose left side is positioned at negative infinity. We apply the functional equation of Riemann zeta function, Euler's reflection formula, and Legendre's duplication formula to evaluate the integral segment through $\Re(z)=-\infty$. After introducing Hurwitz zeta function and properly calculating the difference between the sum of residues in two analogous rectangular contours, we finalize the evaluation. Lastly, we demonstrate the connection of this result with other intricate integrals involving special functions, such as the hyperbolic function. Additionally, we discuss its applications in deriving explicit expressions for the Barnes zeta function.