On a pair of zeta functions

Zhi-Wei Sun

Published 2012-04-30, updated 2015-12-23Version 12

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 $$\zeta_m(1):=\sum_{n=1}^\infty\frac{(-e^{2\pi i/m})^{\omega(n)}}n=0 \ \ \text{and}\ \ \zeta^*_m(1):=\sum_{n=1}^\infty\frac{(-e^{2\pi i/m})^{\Omega(n)}}n=0,$$ and that both $V_m(x)(\log x)^{2\pi i/m}$ and $V_m^*(x)(\log x)^{2\pi i/m}$ have explicit given finite limits as $x\to\infty$, where $$V_m(x)=\sum_{n\le x}\frac{(-e^{2\pi i/m})^{\omega(n)}}n\ \ \text{and}\ \ V_m^*(x)=\sum_{n\le x}\frac{(-e^{2\pi i/m})^{\Omega(n)}}n.$$ We also raise a hypothesis on the parities of $\Omega(n)-n$ which implies the Riemann Hypothesis.