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VI. A “Natural” Proof of the Nonvanishing of L-Series
62
L 7 (z) 1 1 1 1 ,
1 − p −z 1 + ip −z 1 − ip −z 1 + p −z
p≡1 p≡3 p≡7 p≡9
and
1 1 1 1
L 9 (z) .
1 − p −z 1 + p −z 1 + p −z 1 − p −z
p≡1 p≡3 p≡7 p≡9
(Here z> 1 to insure convergence and the subscripting of the
characters is used to reflect the isomorphism of the dual group and
the original group.)
The generating function for the primes in the arithmetic pro-
gressions ((mod 10) in this case) are then linear combinations of
the logarithms of these L-series. And so indeed the crux is the
nonvanishing of these L-series.
What could be more natural or more in the spirit of Dirichlet, but
to prove these separate nonvanishings altogether? So we are led to
take the product of all the L-series! (Landau uses the same device to
prove nonvanishing of the L-series at point 1.)
The result is the Dirichlet series
1 1

Z(z)
−z 4
(1 − p ) (1 − p −4z )
p≡1 p≡3
1 1
× ,
(1 − p −4z ) (1 − p −2z 2
)
p≡7 p≡9
and the problem reduces to showing that Z(z) is zero-free on z 1.
1
Of course, this is equivalent to showing that −z is zero-
p≡1 1−p
free on z 1, which seems, at first glance, to be a more attractive
form of the problem. This is misleading, however, and we are bet-
ter off with Z(z), which is the product of L-series and is an entire
function except possibly for a simple pole at z 1. (See the
appendix.)
Guided by the special cases let us turn to the general one. So let A
be a positive integer, and denote by G A the multiplicative group of
residue classes (mod A) which are prime to A. Set h φ(Að , and
denote the group elements by 1 n 1 ,è 2 ,...,è h . Denote the dual
ˆ arranged
group of G A by G A and its elements by χ 1 ,χ n 2 ,...,χ n h