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<h1 align=center> Zeros on the $\sigma=1$ line
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By the \begin{rawhtml}<a href="../70a">Euler Product</a>\end{rawhtml}, 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 {\it Prime Number Theorem} for $L(s)$.
The name arises as follows.  The classical Prime Number Theorem(PNT):
\begin{eqnarray*}
\pi(x):=&&\sum_{p\le X} 1\cr
\sim &&  \frac{X}{\log X} ,
\end{eqnarray*}
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 [\begin{rawhtml}55 \#5583 \end{rawhtml}]
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 \begin{rawhtml}<a href="../11a">Selberg Class</a>\end{rawhtml}.\begin{rawhtml}
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Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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