Values of zeta-one functions at positive even integers

TL;DR

This paper introduces zeta-one functions based on sums of n^s ± 1, computes their values at positive even integers using residue calculus, and explores their connections to classical number theory theorems.

Contribution

It presents a novel class of zeta-one functions and derives explicit values at positive even integers, expanding the understanding of related special functions.

Findings

Explicit formulas for zeta-one functions at positive even integers

Connections to Euler-Goldbach and Shallit-Zikan theorems

Application of residue theorem in function evaluation

Abstract

Motivated by Euler-Goldbach and Shallit-Zikan theorems, we introduce zeta-one functions with infinite sums of $n^{s}\pm1$ as an analogy of the Riemann zeta function. Then we compute values of these functions at positive even integers by the residue theorem.