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VI
A “Natural” Proof of the

Nonvanishing of L-Series

Rather than the usual adjectives of “elementary” (meaning not in-
volving complex variables) or “simple” (meaning not having too
many steps) which refer to proofs, we introduce a new one, “natural.”
This term, which is just as undefinable as the others, is introduced to
mean not having any ad hoc constructions or brilliancies. A “natural”
proof, then, is one which proves itself, one available to the “common
mathematician in the streets.”
A perfect example of such a proof and one central to our whole
construction is the theorem of Pringsheim and Landau. Here the cru-
cial observation is that a series of positive terms (convergent or not)
can be rearranged at will. Addition remains a commutative operation
when the terms are positive. This is a sum of a set of quantities rather
than the sum of a sequence of them.
The precise statement of the Pringsheim–Landau theorem is that,
for a Dirichlet series with nonnegative coefficients, the real boundary
point of its convergence region must be a singularity.
Indeed this statement proves itself through the observation that
k
n a−z (a−zð k (log n) is a power series in (a − zð with non-
k k!
negative coefficients. Thus the (unique) power series for a n n −z
−a a−z
a n n · n has nonnegative coefficients in powers of (a − zð .
So let b be the real boundary point of the convergence region of
−z
a n n , and suppose that b is a regular point and that b< a. Thus
the power series in (a − zð continues to converge a bit to the left
of b and, by rearranging terms, the Dirichlet series converges there
also, contradicting the meaning of b. A “natural” proof of a “natural”

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