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.

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:

\begin{equation} \zeta(s, l/k) = \frac{k^s}{\varphi (k)} \sum_{\chi \mod k} \bar{\chi} (l) L(s, \chi) \tag{1} \end{equation} \begin{equation} L(s, \chi) = \frac{1}{k^s} \sum_{n = 1}^k \chi(n) \zeta(s, \frac{n}{k}) \tag{2} \end{equation}

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 \begin{equation*} \sum_\chi \bar{\chi}(l) L(s,\chi). \end{equation*} We then expand the L-function and sum over ${\chi}$ first. \begin{equation*} \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}= \end{equation*} \begin{equation*} = \sum_{\substack{ n > 0 \\ n \equiv l \mod k}} \varphi(k) n^{-s}. \end{equation*} In this last line, we used a fact commonly referred to as the Orthogonality of Characters, which says exactly that \begin{equation*} \sum_{\chi \mod k} \bar{\chi}(l) \chi{n} = \begin{cases} \varphi(k) & n \equiv l \mod k \\ 0 & \text{else} \end{cases}. \end{equation*}

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

\begin{equation} \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) \tag{3} \end{equation}

Rearranging the sides, we get that \begin{equation} \zeta(s, l/k) = \frac{k^s}{\varphi(k)} \sum_{\chi \mod k} \bar{\chi}(l) L(s, \chi) \end{equation} 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.^{1}
^{1}This was originally true, but is no longer true. In a
reorganization this post was re-set from the original latex into a format
better suited for the web.

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Comments (1)2013-04-05 anonThis 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).