Dirichlet series for complex powers of the Riemann zeta function

Winston Alarcon-Athens

Published 2021-01-19Version 1

In order to obtain the Dirichlet series for the exponential $\zeta(s)^z$, we define the notion of multiplicative architecture applicable to every integer $ n > 1 $, to which we associate the concept of degree of $n$. These notions are suitable for defining and studying a sequence $(\alpha_n(z))_{n \in {\Bbb Z}^+}$ of polynomials in the indeterminate $z$ which, being used as coefficients of the respective terms of the Dirichlet series of $\zeta(s)$ in the semi-plane $\Re(s)>1$, they convert it into an analitic function $\varphi_s(z)$ that has the characteristic property of exponential functions: $\varphi_s(z_1)\,\varphi_s(z_2) = \varphi_s(z_1 + z_2)$, for all $z_1$, $z_2 \in \Bbb C$. This result allows us to write $\zeta(s)^z = \sum_ {n = 1}^\infty {\alpha_n(z) \over n^s}$ for all $z \in \Bbb C$, since $\varphi_s(1) = \zeta(s) \neq 0$ for all $s\in \Bbb C$ with $\Re(s) > 1$.