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):

where the sum is over the primes , is equivalent to the assertion that when . The deduction of the PNT from the nonvanishing involves applying a Tauberian theorem to . The Tauberian Theorem requires that be regular on , except for the pole at .

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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