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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 a_k is a well-understood arithmetical factor and g_k 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

(Zero-free region, Vinogradov 1958) There is a number c>0 such that there are no zeros of $\zeta(s)$ in the region

$\begin{displaymath}\sigma>1-\frac{c}{(\log(1+\vert t\vert)^{2/3}(\log \log (3+\vert t\vert))^{1/3}}\end{displaymath}$
100% (i.e. almost all) of the zeros of $\zeta(s)$ are in the region $\vert 1/2-\sigma\vert<\phi(t)/\log (1+\vert t\vert)$ for any positive $\phi$ which tends to $\infty$ with t (Selberg 1946).
At least 40% of the zeros of $\zeta(s)$ are on the critical line (Conrey, 1989)

Work on GRH

(Zero-free region Gronwall, 1913, and Titchmarsh 1930) There is a number c>0 such that there are no zeros of $L(s,\chi)$ in the region $\sigma>1-c/\log(1+\vert t\vert q)$ with at most one possible exception. If the exception exists then the character $\chi$ is real and the zero is also real. If such a zero exists, it is called a ``Siegel zero,'' or a ``Landau-Siegel zero.''
The second and third items above hold for Dirichlet L-functions

Work on LH

There is a number $\delta>0$ such that

$\begin{displaymath}\zeta(1/2+it)\ll (1+\vert t\vert)^{1/6-\delta}\end{displaymath}$

The best $\delta$ is about 0.0077 (Bombieri and Iwaniec, 1988, Huxley 1996).

Work on GLH

Burgess (1963):

$\begin{displaymath}L(1/2+it,\chi)\ll q^{3/16+\epsilon}\end{displaymath}$

Work on MH

(Hardy and Littlewood, 1918) g₁=1
(Ingham, 1926) g₂=2.

Work on GMH

(Heath-Brown 1985) The Hardy and Littlewood results hold for characters.
(Huxley 1983) Upper bounds of the correct order of magnitude are given for 6th and 8th moments.

CHRONOLOGY OF EVENTS DURING AIM FUNDING

(Late May) Conrey and Ghosh conjecture g₃=42 for 6th power moment of $\zeta$ , paper written and submitted to IMRN, appeared November
(June) Conrey and Gonek conjecture g₄=24024 for 8th moment of $\zeta$ ; no hope for generalization; paper written, undergoing final revisions
(July) Conrey and Iwaniec prove that

$\begin{displaymath}L(1/2+it,\chi)\ll q^{1/6+\epsilon}\end{displaymath}$

for real characters $\chi$ , beating Burgess bound; paper submitted to Annals of Math.
(July) Iwaniec and Sarnak prove that a hypothesis about primes implies no real zeros (off the critical line) for any L-functions, paper in preparation.
(August) Iwaniec, Sarnak,and Luo show that the low lying zeros of several different families of L-functions obey the statistics predicted by the Katz-Sarnak model
(September) Conrey, Farmer, and Sound prove a result about mollifying mean-values of real characters; paper is ready for submission.
(October) Connes outlines his approach to RH; it builds on what he started in Seattle, but goes well beyond.
(October) Keating and Snaith announce in Vienna that they have a conjecture for all g_k using Random Matrix theory
(October) Balazard and Saias give a condition which implies RH; it is based on Beurling's approach which has been dormant for about 30 years; but it introduces mollifying to the method.
(November) Conrey and Farmer develop the basic properties of the g_k found by Keating and Snaith, including asypmtotics, integrality, and distribution of prime power factors in factorization.
(December) Sound, building on paper with Conrey and Farmer, proves that at least 19/27 of $L(1/2,\chi)$ with $\chi$ real do not vanish; paper in progress uses moments and mollifying.
(December) Conrey and Li prove a counterexample to de Branges approach to RH, which essentially means that de Branges theory developed over the last 12 years is not a viable approach to RH; paper is syubmitted to the Bulletin of the AMS.
(January) Sound improves his result (above) to 7/8, submits paper to Annals.
(February) Conrey and Sound start work on trying to show no real zeros for a positive proportion of moduli.
Conrey and Farmer formulate the general moment for families of L-functions and make a basic conjecture about their values
(May) Conrey and Li prove a formula for the trace of the Hecke operators acting on the space of Maass forms of given eigenvalue and level.
(June) Duke working with Imamoglu works on a Kronecker limit formula for L-functions associated with higher rank automorphic forms.
(July) Conrey and Ramakrishnan show that the L-function associated with any aiutomorphic form over a function field must have zeros on the critical line.
(August) Ramakrishnan working with Rogowski prove a non-vanishing theorem for L-functions associated with modular forms, twisted by real characters, subject to additional constraints on the Fourier coefficients. This work is motivated by Duke's work on $L(1,\chi_d)$ .

Brian Conrey
2000-01-25