On a pair of zeta functions (Q2833098)

The author considers the extensions of well-known functions \(\mu(n)\), \(\lambda(n)\) and \(\zeta(s)\), for \(n \in\mathbb Z_+\) and \(s=\sigma+it\) (the Möbius function, the Liouville function and the Riemann zeta-function, respectively). For \(\sigma>1\) and \(m \in\mathbb Z_+\), let NEWLINE\[NEWLINE \zeta_m(s)=\sum_{n=1}^{\infty}\frac{\big(-e^{2 \pi i/ m}\big)^{\omega(n)}}{n^s} \quad \text{and} \quad \zeta_m^*(s)=\sum_{n=1}^{\infty}\frac{\big(-e^{2 \pi i/ m}\big)^{\Omega(n)}}{n^s}, NEWLINE\]NEWLINE where \(\omega(n)\) and \(\Omega(n)\) denote the number of distinct prime factors of \(n\) and total number of prime factors of \(n\), respectively.NEWLINENEWLINEThese two zeta-functions and related extended arithmetical functions \(\mu_m(n)\), \(\nu_m(n)\) and \(\nu_m^*(n)\) are studied. It is proved that, for \(m>4\), both \(\zeta_m(1)\) and \(\zeta_m^*(1)\) are equal to zero. Also it is shown that, for \(m>4\) and squarefree \(n\), \(\zeta_m(s)=0\).