proving that dirichlet series has non negative coefficients and does not converge for all $s\in\mathbb{C}$

Question

given $Z(s)=\zeta^2(s)\zeta(s+it)\zeta(s-it)$ I need to prove that Z(s) is represented by a dirichlet series with non negative coefficients whiche does not converge for all $s\in\mathbb{C}$.
I have tried to use the dirichlet convolution formula but I didn't know how to continue with what I got.

Answer 1

By the well known Euler product formula for $\zeta(s)$, we have

$$ \log\zeta(s)=\sum_{n\ge1}{a_n\over n^s} $$

where $a_n=\begin{cases}1/k & n=p^k,p\text{ prime} \\ 0 &\text{otherwise}\end{cases}$. This suggests that

\begin{aligned} \log Z(s) &=2\log\zeta(s)+\log\zeta(s+it)+\log\zeta(s-it) \\ &=\sum_{n\ge1}{a_n\over n^s}[2+n^{-it}+n^{it}]=\sum_{n\ge1}{a_n\over n^s}[2+2\cos(t\log n)]:=\sum_{n\ge1}{b_n\over n^s} \end{aligned}

This suggests that the Dirichlet series coefficient for $\log Z(s)$ is nonnegative provided that $t$ is real. Now, using the fact that

$$ e^z=\sum_{n\ge0}{z^n\over n!} $$

we see that the Dirichlet series coefficient for $Z(s)$ is also nonnegative.