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.

About these ads