
\documentclass{article}


\begin{document}

\begin{rawhtml}
<h1 align=center> Zeros of Dirichlet polynomials
</h1>
\end{rawhtml}


Turan showed that if for all sufficiently large $N$, the $N$th partial sum of $\zeta(s)$ does not vanish
 in $\sigma>1$ then the Riemann Hypothesis follows. 

He  [
\begin{rawhtml}
<a href="http://www.ams.org/msnmain?fn=130&s1=10_286a&pg1=MR">MR 10,286a</a>
\end{rawhtml}] strengthened this criterion 
by showing that for every $\epsilon >0$
there is an $N_0(\epsilon)$ such that if the $N$th partial sum
$$\sum_{n=1}^N n^{-s}$$ of the zeta-function has no zeros in 
$\sigma> 1+N^{-1/2+\epsilon}$ for all $N>N_0(\epsilon)$ then the Riemann
Hypothesis holds. 

H. Montgomery [
\begin{rawhtml}
<a href="http://www.ams.org/msnmain?fn=130&s1=87a:11081&pg1=MR">MR 87a:11081</a>
\end{rawhtml}] proved that this approach cannot work because 
for any positive number $c< 4/\pi-1$ 
the $N$th partial sum of $\zeta(s)$ has zeros in the half-plane 
$$\sigma > 1+c (\log \log N) /\log N. $$\begin{rawhtml}
<p> <br>
<hr>
<p> <br>
Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
\end{rawhtml}

\end{document}

