# de Branges' positivity condition

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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