Dirichlet series for complex powers of the Riemann zeta function

TL;DR

This paper introduces a new method to derive Dirichlet series for complex powers of the Riemann zeta function using polynomial sequences, simplifying previous approaches by relying on basic induction and analytic properties.

Contribution

The paper presents a novel, simplified approach to obtain Dirichlet series for complex powers of the Riemann zeta function, avoiding complex relationships with multiplicative functions.

Findings

Derived Dirichlet series for complex powers of zeta function

Introduced polynomial sequences as coefficients in the series

Simplified method using induction and exponential properties

Abstract

To obtain the Dirichlet series for complex powers of the Riemann zeta function, we define and study the basic properties of a sequence of polynomials that, used as coefficients of the respective terms of the Dirichlet series of the Riemann zeta function in the half plane $x > 1$, produces the required exponential function. Unlike the method described in ([4], p.~278), which requires more advanced knowledge of the relationships between Dirichlet series and multiplicative arithmetic functions, our approach only needs mathematical induction on the total number of prime divisors of $n$, the Dirichlet product and the use of an analytic property characteristic of the exponential function in the complex plane.