Dirichlet L-function - Relation To The Hurwitz Zeta-function

Relation To The Hurwitz Zeta-function

The Dirichlet L-functions may be written as a linear combination of the Hurwitz zeta-function at rational values. Fixing an integer k ≥ 1, the Dirichlet L-functions for characters modulo k are linear combinations, with constant coefficients, of the ζ(s,q) where q = m/k and m = 1, 2, ..., k. This means that the Hurwitz zeta-function for rational q has analytic properties that are closely related to the Dirichlet L-functions. Specifically, let χ be a character modulo k. Then we can write its Dirichlet L-function as

$L(s,\chi) = \sum_{n=1}^\infty \frac {\chi(n)}{n^s} = \frac {1}{k^s} \sum_{m=1}^k \chi(m)\; \zeta \left(s,\frac{m}{k}\right).$

In particular, the Dirichlet L-function of the trivial character (which implies the modulus k is prime) yields the Riemann zeta-function:

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