In his 1950 thesis [
MR 12,108g], B. Nyman, a student of A. Beurling, proved that the Riemann Hypothesis is equaivalent to the assertion that
is dense in
Here,
is the space of functions

for which and such that .

Beurling [ MR 17,15a] proved that the following three statements regarding a number with are equivalent:

(1) has nozeros in

(2) is dense in (3) The characteristic function is in the closure of in

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