Zeta functions for Riemann zeros

TL;DR

This paper introduces a family of Zeta functions constructed from Riemann zeros, demonstrating their meromorphic extension across the complex plane and detailing their analytical properties and special values.

Contribution

It provides explicit meromorphic extensions and analytical features of Zeta functions built from Riemann zeros, advancing understanding of their complex structure.

Findings

Zeta functions have meromorphic extensions over the entire complex plane.

Explicit descriptions of polar structures and special values are provided.

The functions exhibit countably many special values with analytical significance.

Abstract

A family of Zeta functions built as Dirichlet series over the Riemann zeros are shown to have meromorphic extensions in the whole complex plane, for which numerous analytical features (the polar structure, plus countably many special values) are explicitly displayed.