By the Euler Product, the -function
does
not vanish in the half-plane
. Thus, the simplest
nontrivial assertion about the zeros of
is that
does not vanish on the
line. Such a result
is known as a Prime Number Theorem for
.
The name arises as follows. The classical Prime Number Theorem(PNT):
The Prime Number Theorem for was proven by Hadamard
and de la Valee Poissin in 1896. Jacquet and Shalika [55 #5583]
proved the corresponding result for
-functions associated to
automorphic representations on
. It would be significant
to prove such a result for the Selberg Class.
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for The Riemann Hypothesis.