# §25.15 Dirichlet -functions

## §25.15(i) Definitions and Basic Properties

The notation was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series

25.15.1,

where is a Dirichlet character 27.8). For the principal character , is analytic everywhere except for a simple pole at with residue , where is Euler’s totient function (§27.2). If , then is an entire function of .

with the product taken over all primes , beginning with . This implies that if .

Equations (25.15.3) and (25.15.4) hold for all if , and for all () if :

where is a primitive character (mod ) for some positive divisor of 27.8).

When is a primitive character (mod ) the -functions satisfy the functional equation:

where is the complex conjugate of , and

## §25.15(ii) Zeros

Since if , (25.15.5) shows that for a primitive character the only zeros of for (the so-called trivial zeros) are as follows:

There are also infinitely many zeros in the critical strip , located symmetrically about the critical line , but not necessarily symmetrically about the real axis.

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