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