§25.15 Dirichlet $L$-functions

The notation $L(s,\chi )$ was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series

where $\chi (n)$ is a Dirichlet character $(modk)$ (§27.8). For the principal character ${\chi }_1(modk)$, $L(s,{\chi }_1)$ is analytic everywhere except for a simple pole at $s=1$ with residue $\varphi \left(k\right)/k$, where $\varphi \left(k\right)$ is Euler’s totient function (§27.2). If $\chi \ne {\chi }_1$, then $L(s,\chi )$ is an entire function of $s$.

with the product taken over all primes $p$, beginning with $p=2$. This implies that $L(s,\chi )\ne 0$ if $\mathrm{\Re }s>1$.

Equations (25.15.3) and (25.15.4) hold for all $s$ if $\chi \ne {\chi }_1$, and for all $s$ ($\ne 1$) if $\chi ={\chi }_1$:

where ${\chi }_0$ is a primitive character (mod $d$) for some positive divisor $d$ of $k$ (§27.8).

When $\chi$ is a primitive character (mod $k$) the $L$-functions satisfy the functional equation:

where $\overline{\chi }$ is the complex conjugate of $\chi$, and

Since $L(s,\chi )\ne 0$ if $\mathrm{\Re }s>1$, (25.15.5) shows that for a primitive character $\chi$ the only zeros of $L(s,\chi )$ for (the so-called trivial zeros) are as follows:

There are also infinitely many zeros in the critical strip $0\le \mathrm{\Re }s\le 1$, located symmetrically about the critical line $\mathrm{\Re }s=\frac{1}{2}$, but not necessarily symmetrically about the real axis.

where ${\chi }_1$ is the principal character $(modk)$. This result plays an important role in the proof of Dirichlet’s theorem on primes in arithmetic progressions (§27.11). Related results are: