Page 68 -

VI. A “Natural” Proof of the Nonvanishing of L-Series
is an isomorphism of G and G. Next, for z>
so that n i ↔ χ n i
1 ˆ 63
(z)
1, write L n i −z and finally set Z(z)
n j p≡n j 1−χ n i (n j )p

(z). As in the case A 10, elementary algebra leads to
L n i
n i
1
Z(z) , where h j is the order of the group
n j p≡n j (1−p −h j z h/h j
)
element n j .
As before, Z(z) is entire except possibly for a simple pole at z 1,
andweseekaproofthatZ(1+ia) ø 0 forreala.Soagainweassume
2
Z(1 + ia) 0, form Z (z)Z(z + ia)Z(z − ia), and conclude that
it is entire. We note that its logarithm and hence that it itself has
nonnegative coefficients so that (1) is applicable.
So, with dazzling speed, we see that a zero of any L-series would
lead to the everywhere convergence of the Dirichlet series (with
2
nonnegative coefficients) Z (z)Z(z + ia)Z(z − ia).
The end game (final contradiction) is also as before although 2
may not be among the primes in the resultant product, and we may
have to take some other prime π. Nonetheless again we see that the
subseries of powers of π diverges at z 0 which gives us our QED.

Appendix. A proof that the L-series are everywhere analytic func-
tions with the exception of the principal L-series, L 1 at the single
point z 1, which is a simple pole.

∞ 1 1
Lemma. For any θ in [0,1), define fàz) z − for
n 1 (n−θ) z−1
z> 1. Then fàz) is continuable to an entire function.
∞ −nt θt z−1 1 ∞ −t
Proof. Since, for z> 1, e e t dt e ×
0 (n−θ) z 0
t z−1 dt A(z) z , by summing, we get
(n−θ)

θt
1 1 ∞ e z−1
× t dt
t
(n − θ) z A(z) 0 e − 1
or

∞ θt −t
1 1 1 e e z−1
− − t dt.
t
(n − θ) z z − 1 A(z) 0 e − 1 t