# Relation between zeta function and Dirichlet L-function

Let $$H(s)=\frac{\zeta(s)}{\phi(q)} \sum_{\chi \mod{q}} L(s,\chi)=\sum_{n=1}^{\infty} \frac{h(n)}{n^s}$$ What is the smallest n (as a function of q) such that $h(n)\neq 1$?

-

## 1 Answer

The orthogonality relations for Dirichlet characters imply that$$\frac1{\phi(q)} \sum_{\chi\pmod q} L(s,\chi) = \sum_{n\equiv1\pmod q} n^{-s}.$$ Therefore$$H(s) = \zeta(s) \sum_{n\equiv1\pmod q} n^{-s} = \sum_{n=1}^\infty n^{-s} \sum_{\substack{d\mid n \\ d\equiv1\pmod q}} 1,$$ from which you can answer your particular question.

-