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VI. A “Natural” Proof of the Nonvanishing of L-Series
2
(2) ζ (z)ζ(z +ia)ζ(z −ia) is an entire Dirichlet series with nonneg-
ative coefficients. Combining this with (1) implies the unbelievable
fact that
2
(3) the Dirichlet series for ζ (z)ζ(z + ia)ζ(z − ia) is everywhere
convergent.
The falsity of (3) can be established in may ways, especially if
we recall that the coefficients are all nonnegative. For example, the
subseries corresponding to n power of 2 is exactly equal to
1 1 1 1 1 along the
−z 2 ·
−z 2 ·
(1−2 ) 1−2 −z−ia · 1−2 −z+ia which exceeds (1−2 ) 4
nonnegative (real) axis and thereby guarantees divergence at z
0. Q.E.D.
And so we have the promised natural proof of the nonvanishing of
the ζ-function which can then lead to the natural proof of the Prime
Number Theorem. We must turn to the general L-series which holds
the germ of the proof of the Prime Progression Theorem. Dirichlet
pointed out that the natural way to treat these progressions is not
one progression at a time but all of the pertinent progressions of a
given modulus simultaneously, for this leads to the underlying group
and hence to its dual group, the group of characters. Let us look, for
example, at the modulus 10. The pertinent progressions are 10k + 1,
10k + 3, 10k + 7,10k + 9, so that the group is the multiplicative
group of 1,3,7,9 (mod 10). The characters are

χ 1 : χ 1 (1) 1,χ 1 (3) 1,χ 1 (7) 1,χ 1 (9) 1,
χ 3 : χ 3 (1) 1,χ 3 (3) 1,χ 3 (7) 1,χ 3 (9) 1,

χ 7 : χ 7 (1) 1,χ 7 (3) 1,χ 7 (7) 1,χ 7 (9) 1,

χ 9 : χ 9 (1) 1,χ 9 (3) 1,χ 9 (7) 1,χ 9 (9) 1,

and so the L-series are

1 1 1 1

L 1 (z) ,
1 − p −z 1 − p −z 1 − p −z 1 − p −z
p≡1 p≡3 p≡7 p≡9
1 1 1 1
L 3 (z) ,
1 − p −z 1 − ip −z 1 + ip −z 1 + p −z
p≡1 p≡3 p≡7 p≡9