Zeros on the $\sigma=1$ line

By the Euler Product, the $L$-function $L(s)$ does not vanish in the half-plane $\sigma>1$. Thus, the simplest nontrivial assertion about the zeros of $L(s)$ is that $L(s)$ does not vanish on the $\sigma=1$ line. Such a result is known as a Prime Number Theorem for $L(s)$. The name arises as follows. The classical Prime Number Theorem(PNT):

\pi(x):=&&\sum_{p\le X} 1\cr
\sim && \frac{X}{\log X} ,

where the sum is over the primes $p$, is equivalent to the assertion that $\zeta(s)\not=0$ when $\sigma=1$. The deduction of the PNT from the nonvanishing involves applying a Tauberian theorem to $\zeta'/\zeta$. The Tauberian Theorem requires that $\zeta'/\zeta$ be regular on $\sigma=1$, except for the pole at $s=1$.

The Prime Number Theorem for $\zeta(s)$ was proven by Hadamard and de la Valee Poissin in 1896. Jacquet and Shalika [55 #5583] proved the corresponding result for $L$-functions associated to automorphic representations on $GL(n)$. It would be significant to prove such a result for the Selberg Class.

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