THE SIEGEL ZERO PROJECT: BACKGROUND AND CHRONOLOGY
Some Definitions. The Riemann Hypothesis (RH) is the assertion that all critical zeros of the Riemann $\zeta$-function are on the critical line. The Grand Riemann Hypothesis is the assertion that all critical zeros of Dirichlet L-functions are on the critical line. A consequence of RH is the Lindelöf Hypothesis (LH) which is a bound for $\zeta$ on the 1/2-line, namely

\begin{displaymath}\zeta(1/2+it)\ll (1+\vert t\vert)^\epsilon\end{displaymath}

for any fixed $\epsilon>0$. GLH is the assertion that

\begin{displaymath}L(1/2+it,\chi) \ll (1+q\vert t\vert)^\epsilon\end{displaymath}

for any $\epsilon>0$ where $\chi$ is a Dirichlet character mod q. The Moment Hypothesis (MH) is that

\begin{displaymath}\int_0^T\vert\zeta(1/2+it)\vert^{2k}~dt\sim \frac{g_k a_k}{\Gamma(1+k^2)} T(\log T)^{k^2}\end{displaymath}

where ak is a well-understood arithmetical factor and gk is a mysterious geometric factor. GMH is that

\begin{displaymath}\sum_{q\le Q}\frac{1}{q}\sum_{\chi \mbox{ mod }q}^* \vert L(1...
...)\vert^{2k}\sim \frac{g_k a_k}{\Gamma(1+k^2)} Q (\log Q)^{k^2} \end{displaymath}

where the * indicates that the sum is over primitive characters. GMH is related to both of GRH and GLH but is implied by neither. Some of the techniques used to approach GLH are moment techniques and to approach GRH are moment techniques with mollifiers.


WHAT IS KNOWN?
Results toward RH Work on GRH Work on LH Work on GLH Work on MH Work on GMH


CHRONOLOGY OF EVENTS DURING AIM FUNDING




 
Brian Conrey
2000-01-25