Hurwitz Zeta is a sum of Dirichlet L Functions, and vice-versa

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

Lemma 1

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

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

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

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

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

$latex \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}= $

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

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

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

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

Rearranging the sides, we get that

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

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

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.

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One Response to Hurwitz Zeta is a sum of Dirichlet L Functions, and vice-versa

  1. anon says:

    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).

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