# 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