On a pair of zeta functions

TL;DR

This paper introduces and analyzes two new zeta functions involving prime factor counts, revealing identities and conjectures related to prime number theory and the Riemann Hypothesis.

Contribution

It defines two novel zeta functions based on prime factorization counts and proves their vanishing at s=1 for certain parameters, linking to prime number theorems and hypothesizing a connection to the Riemann Hypothesis.

Findings

Sum over squarefree n of (-e^{2πi/m})^{ω(n)}/n equals zero for m>4

Both zeta functions vanish at s=1 for m>4

Proposes a hypothesis on prime factor parity implying the Riemann Hypothesis

Abstract

Let $m$ be a positive integer, and define $$\zeta_m(s)=\sum_{n=1}^\infty\frac{(-e^{2\pi i/m})^{\omega(n)}}{n^s}\ \ \ \ \text{and} \ \ \ \ \zeta^*_m(s)=\sum_{n=1}^\infty\frac{(-e^{2\pi i/m})^{\Omega(n)}}{n^s},$$ for $\Re(s)>1$, where $\omega(n)$ denotes the number of distinct prime factors of $n$, and $\Omega(n)$ represents the total number of prime factors of $n$ (counted with multiplicity). In this paper we study these two zeta functions and related arithmetical functions. We show that $$\sum^\infty_{n=1\atop n\ \text{is squarefree}}\frac{(-e^{2\pi i/m})^{\omega(n)}}n=0\quad\text{if}\ \ m>4,$$ which is similar to the known identity $\sum_{n=1}^\infty\mu(n)/n=0$ equivalent to the Prime Number Theorem. For $m>4$, we prove that $$\zeta_m(1):=\sum_{n=1}^\infty\frac{(-e^{2\pi i/m})^{\omega(n)}}n=0 \ \ \ \ \text{and}\ \ \ \ \zeta^*_m(1):=\sum_{n=1}^\infty\frac{(-e^{2\pi i/m})^{\Omega(n)}}n=0.$$ We also raise a hypothesis on the parities of $\Omega(n)-n$ which implies the Riemann Hypothesis.