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 ${s}$ and real ${0 < a \leq 1}$ is ${\zeta(s,a) := \displaystyle \sum_{n = 0}^\infty \frac{1}{(n + a)^s}}$. A Dirichlet L-function is a function ${L(s, \chi) = \displaystyle \sum_{n = 1}^\infty \frac{\chi (n)}{n^s}}$, where ${\chi}$ is a Dirichlet character. This note contains a few proofs of the following relations:

Lemma 1

$\displaystyle \zeta(s, l/k) = \frac{k^s}{\varphi (k)} \sum_{\chi \mod k} \bar{\chi} (l) L(s, \chi) \ \ \ \ \ (1)$

$\displaystyle L(s, \chi) = \frac{1}{k^s} \sum_{n = 1}^k \chi(n) \zeta(s, \frac{n}{k}) \ \ \ \ \ (2)$

Proof: We start by considering ${L(s, \chi)}$ for a Dirichlet Character ${\chi \mod k}$. We multiply by ${\bar{\chi}(l)}$ for some ${l}$ that is relatively prime to ${k}$ and sum over the different ${\chi \mod k}$ to get

$\displaystyle \sum_\chi \bar{\chi}(l) L(s,\chi)$

We then expand the L-function and sum over ${\chi}$ first.

$\displaystyle \sum_\chi \bar{\chi}(l) L(s,\chi)= \sum_\chi \bar{\chi} (l) \sum_n \frac{\chi(n)}{n^s} = \sum_n \sum_\chi \left( \bar{\chi}(l) \chi(n) \right) n^{-s}=$

$\displaystyle = \sum_{\substack{ n > 0 \\ n \equiv l \mod k}} \varphi(k) n^{-s}$

In this last line, we used a fact commonly referred to as the “Orthogonality of Characters” , which says exactly that ${\displaystyle \sum_{\chi \mod k} \bar{\chi}(l) \chi{n} = \begin{cases} \varphi(k) & n \equiv l \mod k \\ 0 & \text{else} \end{cases}}$.

What are the values of ${n > 0, n \equiv l \mod k}$? They start ${l, k + l, 2k+l, \ldots}$. If we were to factor out a ${k}$, we would get ${l/k, 1 + l/k, 2 + l/k, \ldots}$. So we continue to get

$\displaystyle = \sum_{\substack{ n > 0 \\ n \equiv l \mod k}} \varphi(k) n^{-s} = \varphi(k) \sum_n \frac{1}{k^s} \frac{1}{(n + l/k)^s} = \frac{\varphi(k)}{k^s} \zeta(s, l/k) \ \ \ \ \ (3)$

Rearranging the sides, we get that

$\displaystyle \zeta(s, l/k) = \frac{k^s}{\varphi(k)} \sum_{\chi \mod k} \bar{\chi}(l) L(s, \chi)$

To write ${L(s,\chi)}$ as a sum of Hurwitz zeta functions, we multiply by ${\chi(l)}$ and sum across ${l}$. Since ${\chi(l) \bar{\chi}(l) = 1}$, the sum on the right disappears, yielding a factor of ${\varphi(k)}$ since there are ${\varphi(k)}$ characters ${\mod k}$. $\Box$

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.