Let be an entire function satisfying
for
in the upper half-plane. Define a Hilbert space of entire functions
to be the set of all entire functions such that
is square integrable on the real axis and such that

for all complex , where the inner product of the space is given by

for all elements and where

is the reproducing kernel function of the space , that is, the identity

holds for every complex and for every element .

de Branges [ MR 87m:11050] and [ MR 93f:46032] proved the following beautiful

**Theorem .**
*Let be an entire function having no real zeros
such that
for , such that
for a constant of absolute value one, and such that
is a strictly increasing function of for each fixed real .
If
for every
element
with
, then the zeros
of lie on the line , and
when is a zero of . *

Let
. Then it can be shown that
for , and that
is a strictly increasing function of on for
each fixed real .
Therefore, it is natural to ask whether the Hilbert
space of entire functions satisfies the condition that

for every element of such that , because if so, then the Riemann Hypothesis would follow.

It is shown in [ MR 2001h:11114] that this condition is not satisfied.

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