On a pair of zeta functions

Zhi-Wei Sun

Let $m$ be a positive integer, and define $$ζ_m(s)=\sum_{n=1}^\infty\frac{(-e^{2πi/m})^{ω(n)}}{n^s}\ \ \ \ \text{and} \ \ \ \ ζ^*_m(s)=\sum_{n=1}^\infty\frac{(-e^{2πi/m})^{Ω(n)}}{n…