An incomplete Riemann Zeta function as a fractional integral

TL;DR

This paper presents a novel representation of the incomplete Riemann zeta function as a fractional integral, revealing unique properties and potential for new developments in understanding the zeta function.

Contribution

It introduces a fractional integral formulation of the incomplete Riemann zeta function, establishing new properties and relationships not previously known.

Findings

The incomplete Riemann zeta function can be expressed as a lower-bounded Riemann-Liouville fractional integral.

This representation obeys the semigroup property of fractional integrals.

It allows for additional functional relationships through differentiation in certain regions.

Abstract

An incomplete Riemann zeta function can be expressed as a lower-bounded, improper Riemann-Liouville fractional integral, which, when evaluated at $0$, is equivalent to the complete Riemann zeta function. Solutions to Landau's problem with $\zeta(s) = \eta(s)/0$ establish a functional relationship between the Riemann zeta function and the Dirichlet eta function, which can be represented as an integral for the positive complex half-plane, excluding the pole at $s = 1$. This integral can be related to a lower-bounded Riemann-Liouville fractional integral directly via Cauchy's Formula for repeated integration extended to the complex plane with improper bounds. In order to establish this relationship, however, specific existence conditions must be met. The incomplete Riemann zeta function as a fractional integral has some unique properties that other representations lack: First, it obeys the semigroup property of fractional integrals; second, it allows for an additional functional relationship to itself through differentiation in other regions of convergence for its fractional integral representation. The authors suggest development of the Riemann zeta function using this representation and its properties.