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§25.15 Dirichlet L-functions

Contents

§25.15(i) Definitions and Basic Properties

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

25.15.1 L(s,χ)=n=1χ(n)ns,
s>1,

where χ(n) is a Dirichlet character (modk)27.8). For the principal character χ1(modk), L(s,χ1) is analytic everywhere except for a simple pole at s=1 with residue ϕ(k)/k, where ϕ(k) is Euler’s totient function (§27.2). If χχ1, then L(s,χ) is an entire function of s.

25.15.2 L(s,χ)=p(1-χ(p)ps)-1,
s>1,

with the product taken over all primes p, beginning with p=2. This implies that L(s,χ)0 if s>1.

Equations (25.15.3) and (25.15.4) hold for all s if χχ1, and for all s (1) if χ=χ1:

25.15.3 L(s,χ) =k-sr=1k-1χ(r)ζ(s,rk),
25.15.4 L(s,χ) =L(s,χ0)p|k(1-χ0(p)ps),

where χ0 is a primitive character (mod d) for some positive divisor d of k27.8).

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

25.15.5 L(1-s,χ)=ks-1Γ(s)(2π)s(-πs/2+χ(-1)πs/2)G(χ)L(s,χ¯),

where χ¯ is the complex conjugate of χ, and

25.15.6 G(χ)=r=1kχ(r)2πr/k.

§25.15(ii) Zeros

Since L(s,χ)0 if s>1, (25.15.5) shows that for a primitive character χ the only zeros of L(s,χ) for s<0 (the so-called trivial zeros) are as follows:

25.15.7 L(-2n,χ)=0 if χ(-1)=1,
n=0,1,2,,
25.15.8 L(-2n-1,χ)=0 if χ(-1)=-1,
n=0,1,2,.

There are also infinitely many zeros in the critical strip 0s1, located symmetrically about the critical line s=12, but not necessarily symmetrically about the real axis.

25.15.9 L(1,χ)0 if χχ1,

where χ1 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:

25.15.10 L(0,χ)={-1kr=1krχ(r),χχ1,0,χ=χ1.