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VI. A “Natural” Proof of the Nonvanishing of L-Series
60
theorem follows, one with a very nice corollary which we record for
future use.
(1) If a Dirichlet series with nonnegative coefficients represents a
function whichis (canbecontinuedtobe)entire, thenit iseverywhere
convergent.
Our ultimate aim is to prove that the L-series have no zeros on
the line z 1. This is the nonvanishing of the L-series that we
referred to in the chapter title. So let us begin with the simplest of
1
all L-series, the ζ-function, ζ(z) z . Our proof, in fact, was
n
noticed by Narasimhan and is as follows: Assume, par contraire, that
ζ(z) had a zero at 1+ia, a real. Then (sic!) the function ζ(z)ζ(z+ia)
would be entire. (See the appendix, page no. 63).
The only trouble points could be at z 1orat z 1 − ia where
one of the factors has a pole, but these are then cancelled by the other
factor, which, by our assumption, has a zero.
A bizarre conclusion, perhaps, that the Dirichlet series ζ(z)ζ(z +
ia) is entire. But how to get a contradiction? Surely there is no hint
from its coefficients, they aren’t even real. A natural step then would
be to make them real by multiplying by the conjugate coefficient
function, ζ(z)ζ(z − ia), which of course is also entire. We are led,
2
then, to form ζ (z)ζ(z + ia)ζ(z − ia).
This function is entire and has real coefficients, but are they pos-
itive? (We want them to be so that we can use (1).) Since these are
complicated coefficients dependent on sums of complex powers of
divisiors, we pass to the logarithm, 2 log ζ(z) + log ζ(z + ia) +
log ζ(z − ia), which, by Euler’s factorization of the ζ-function, has
simple coefficients. A dangerous route, passing to the logarithm, be-
cause this surely destroys our everywhere analyticity. Nevertheless
let us brazen forth (faint heart fair maiden never won).
By Euler’s factorization, 2 log ζ(z) + log ζ(z + ia) + log ζ(z −
1 1 1 1
ia) 2 log −z + log −z−ia + log
p 1−p 1−p 1−p −z+ia p,v vp vz
(2 + p −iva + p +iva ), and indeed these coefficients are nonnega-
tive! The dangerous route is now reversed by exponentiating. We
return to our entire function while preserving the nonnegativity of
the coefficients. All in all, then,