Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 23 / 4 / b / Solution

The Dirichlet-series coefficients areThey are multiplicative. At a prime power, their values are $k+1$ if ${\chi }_D(p)=1$, one for even $k$ and zero for odd $k$ if ${\chi }_D(p)=-1$, and one if ${\chi }_D(p)=0$. Thus every $a_n$ is nonnegative. In particular $a_{m^2}\ge 1$. This is the nonnegative zeta-times-real-L coefficients identity. The same Euler expansion gives $-{\zeta }^{\prime }/\zeta (\sigma )-L^{\prime }/L(\sigma ,{\chi }_D)\ge 0$ for $\sigma >1$, with coefficients $\Lambda (n)(1+{\chi }_D(n))\ge 0$.