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.

The Hurwitz zeta function, for complex and real is . A Dirichlet L-function is a function , where is a Dirichlet character. This note contains a few proofs of the following relations:

**Lemma 1**

*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.

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*Related*

This is a corollary to the fact that the characters of a group representation form a basis for the space of class functions; it allows “fourier” decomposition and inversion of such functions. On an abelian group – here the unit groups of integers mod k – the conjugacy classes are the singletons, so we may work without restriction on the functions. It is important n/k is in simplest terms so that n resides in U(k) and we can consider the corresponding indicator function (of the equivalence class of given residue n).