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 1This 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.
Leave a comment
Info on how to comment
To make a comment, please send an email using the button below. Your email address won't be shared (unless you include it in the body of your comment). If you don't want your real name to be used next to your comment, please specify the name you would like to use. If you want your name to link to a particular url, include that as well.
bold, italics, and plain text are allowed in
comments. A reasonable subset of markdown is supported, including lists,
links, and fenced code blocks. In addition, math can be formatted using
$(inline math)$ or $$(your display equation)$$.
Please use plaintext email when commenting. See Plaintext Email and Comments on this site for more. Note also that comments are expected to be open, considerate, and respectful.
Comments (1)
2013-04-05 anon
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).