A modified M\"obius $\mu$-function

TL;DR

This paper studies a modified Möbius μ-function connected to an infinite product of shifted Riemann zeta-functions, providing bounds for its summatory function and exploring its relation to the Riemann hypothesis.

Contribution

It introduces a modified Möbius μ-function linked to shifted zeta-functions and establishes bounds for its summatory function, advancing understanding of its properties and connections to the Riemann hypothesis.

Findings

Established conditional and unconditional bounds for the summatory function

Analyzed the relation between the modified μ-function and the Riemann hypothesis

Discussed implications for number theory and the distribution of primes

Abstract

We investigate a modified M\"obius $\mu$-function which is related to an infinite product of shifted Riemann zeta-functions. We prove conditional and unconditional upper and lower bounds for its summatory function, and, finally, we discuss relations with Riemann's hypothesis.