# Is it possible to improve on Siegel's theorem for exceptional zeroes?

Let $\chi$ be a real nonprincipal character modulo $q$. Siegel's theorem on exceptional zeroes states that for any $\epsilon >0$ there exists a positive number $C(\epsilon)$ such that, for any real zero $\beta$ of $L(s,\chi)$,

$\displaystyle\beta \leqslant 1-\frac{C(\epsilon)}{q^\epsilon}.$

This is a superior estimate to anything that can currently be said about exceptional zeroes (though it is ineffective, in the sense that the method of proof makes it impossible to assign a numerical value to $C(\epsilon)$ for given $\epsilon$). Regardless, it is still very useful in applications; for example to put a bound on $\psi(x,\chi)$ for primitive $\chi$, which one needs to prove the Bombieri-Vinogradov theorem. Improving Siegel's estimate may even pave a way to a proof of the (weak) Elliott-Halberstam conjecture, if a few other theoretical leaps are made.

Question: What is the next logical step (if any) after this theorem -- is it true that the only way we can move forward is to disprove the existence of Siegel zeroes?

Also, what is a 'reasonable' conjecture to make? Is there any way to improve on Siegel's estimate?

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Improving on Siegel's theorem is a major open problem in analytic number theory. –  Matt Young Mar 12 '11 at 19:11

As Matt's comment indicates, this is a very broad question. One possible 'way to move forward' is considered in the paper of Sarnak and Zaharescu titled Some remarks on Landau-Siegel zeros, Duke Math. J. v. 111 (2002), pp. 495–507. They show the surprising (to me) result that if the only zeros off the critical line are real (ie allowing Landau-Siegel zeros) one can still get good lower bounds on the class number.

A general survey of the problem of the Siegel zero, discussing the Sarnak-Zarahescu result above and many other interesting developments, can be found in Iwaniec's Conversations on the Exceptional Character