The first use of the letter L to denote these functions was by Dirichlet in 1837 (see *Werke I* [MR:249268] pages 313-342)
who
used L-functions to prove that there are infinitely many primes in any (primitive) arithmetic progression.
He considered
series of the form
$$L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s}$$
where $\chi$ is a (Dirichlet) character, which is the extension to the integers of a character of the
group $(\mathbb Z/q\mathbb Z)^*$. For example the arithmetic function $\chi$ defined by

\[\chi(n)=\begin{cases} 1 & \textrm{ if } n\equiv 1 \bmod 3\\ -1 & \textrm{ if } n\equiv 2\bmod 3\\ 0 & \textrm{ if }n\equiv 0\bmod 3\end{cases}\]

is a Dirichlet character modulo 3. The above L-function has an Euler product
$$L(s,\chi)=\prod_{p }\left(1-\frac{\chi(p)}{p^s}\right)^{-1}$$
and satisfies a functional equation that
$$\left(\frac{3}{\pi}\right)^{\frac{s}{2}}\Gamma\left(\frac{s+1}{2}\right) L(s,\chi)$$
is invariant under $s\to 1-s$. Also, note the special value
$$L(1,\chi)= \frac{\pi}{3\sqrt{3}}; $$
Dirichlet needed to know that his L-functions did not vanish at 1 and he
used special values to prove this fact.
Dirichlet's original proof was for prime moduli; for composite moduli he required
his class number formula which was proven in 1839 - 1840 (see his *Werke I*, pp. 411-496).

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- Last edited by Stephan Ehlen on 2019-05-03 08:29:35

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