At least three times now, I have needed to use that Hurwitz Zeta functions are a sum of L-functions and its converse, only to have forgotten how it goes. And unfortunately, the current wikipedia article on the Hurwitz Zeta function has a mistake, omitting the $varphi$ term (although it will soon be corrected). Instead of re-doing it each time, I write this detail here, below the fold.
Proof: We start by considering for a Dirichlet Character . We multiply by for some that is relatively prime to and sum over the different to get
We then expand the L-function and sum over first.
In this last line, we used a fact commonly referred to as the “Orthogonality of Characters” , which says exactly that .
What are the values of ? They start . If we were to factor out a , we would get . So we continue to get
Rearranging the sides, we get that
To write as a sum of Hurwitz zeta functions, we multiply by and sum across . Since , the sum on the right disappears, yielding a factor of since there are characters .
I’d like to end that the exact same idea can be used to first show that an L-function is a sum of Hurwitz zeta functions and to then conclude the converse using the heart of the idea for of equation 3.
Further, this document was typed up using latex2wp, which I cannot recommend highly enough.