Dirichlet L-Function from Trivial Character

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Theorem

Let $\chi_0$ be the trivial Dirichlet character modulo $q$.



Let $\zeta$ be the Riemann zeta function.

Then:

$\displaystyle \map L {s, \chi_0} = \map \zeta s \cdot \prod_{p \mathop \divides q} \paren {1 - p^{-s} }$

where $\divides$ denotes divisibility.


Proof

By definition:

$\map {\chi_0} a = \begin{cases} 1 & : \gcd \set {a, q} = 1 \\ 0 & : \text{otherwise} \end{cases}$



Therefore:

\(\ds \map L {s, \chi_0}\) \(=\) \(\ds \prod_p \paren {1 - \map \chi p p^{-s} }^{-1}\) Euler Product: $p$ ranges over the primes
\(\ds \) \(=\) \(\ds \prod_{p \mathop \nmid q} \paren {1 - p^{-s} }^{-1}\) Fundamental Theorem of Arithmetic: $\gcd \set {p, q} > 1$ if and only if $p \divides q$
\(\ds \) \(=\) \(\ds \prod_{p \mathop \divides q} \paren {1 - p^{-s} } \prod_p \paren {1 - p^{-s} }^{-1}\)
\(\ds \) \(=\) \(\ds \map \zeta s \prod_{p \mathop \divides q} \paren {1 - p^{-s} }\) Euler Product



Hence the result.

$\blacksquare$