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The short version of my question goes: What is known to follow from the existence of Siegel zeros?

A longer version to give an idea of what I have in mind: The "expectional zeros" of course first cropped up in the work of Deuring and Heilbronn on the Gauss class number problem, the finitude of discriminants $D$ where $h(D) = 1$ (or $h(D) = c$ in general) is shown to follow from the failure of the RH (Deuring-Mordell) or, more generally, the failure of the GRH (Heilbronn). Since the finitude also follows (with good bounds) on the assumption of the GRH (Hecke-Landau) this settles the result unconditionally. (A more careful analysis by Heilbronn-Linfoot gives explicit bounds so that $h(D) = 1$ can happen at most 10 times.) Siegel later crystallised out the peculiar case of the exceptional real zero and proved the result in a very sharp form.

Linnik then used these ideas to show that $P(a,q) \ll q^L$, $P(a,q)$ being the least prime $\equiv a \pmod q$ and $L$ an explicit constant. Again this was known to follow on the GRH (with $L < 2 + \varepsilon$), hence the result is unconditional.

Iwaniec [Conversations on the Exceptional Character, p. 118 ] mentions on the other hand that Siegel zeros can be exploited to give even stronger results than what is available on the (presumably true) GRH; for example it follows that $L < 2$, hence improving on the bound known on the GRH. This is related to earlier work by Heath-Brown showing that the existence of exceptional zeros implies both that $L < 3$ (which is better than what is known unconditionally, although not better than the $L < 2 + \varepsilon$ that follows on the GRH) and the existence of infinitely many twin primes.

The results of Heath-Brown and Iwaniec, however, assumes an infinite number of exceptional zeros, to increasing modulus $q$, to derive results stronger than what is known unconditionally (or even on the GRH). What I would like to know is if similar results are proved in the literature working only from a single exceptional zero (or even from the weaker assumption of the mere existence of any zeros off the critical line, like Deuring and Heilbronn did)? Putting $\Theta =$ supremum of the real parts of the zeros, we have of course that $1/2 < \Theta$ (i.e. the failure of the RH) would mess up the approximate formulas for different prime counting functions (e.g. by a theorem of Schmidt we have $\Pi(x) - {\rm li} (x) = \Omega_\pm(x^{\Theta-\varepsilon})$, thus $= \Omega_\pm(x^{1/2})$ in case RH fails), but as long as the mere existence of any zero off the line is assumed (not some horrifying disaster like $\Theta = 1$), I know of no results stronger than the unconditionally known irregularites in the approximation (e.g. from Littlewood's theorem). The assumption of a Siegel zero of course is an assumption of such a special failure of the GRH on the other hand, so perhaps I'm missing some rather obvious examples?

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@Kálmán: I am not sure if you understood what kiskis said below. Heath-Brown proved that the existence of an exceptional zero implies the twin prime conjecture: Prime twins and Siegel zeros, Proc. London Math. Soc. (3) 47 (1983), no. 2, 193-224. –  GH from MO May 19 '13 at 21:44
    
@GH: I mentioned this result in my question above, but "an" exceptional zero isn't quite right: H-B crucially relies on an infinitude of them. What kiskis was hinting at was a possible unconditional proof of the twin prime conjecture though (unless I misunderstood), to use the result of H-B for that you need to show that the conjecture holds also conditioned on the absence of (an infinitude of) Siegel zeros. –  Kálmán Kőszegi May 19 '13 at 21:59
    
@Kálmán: The notion of exceptional zero does not make sense for a single zero (this is also what Terry Tao tried to say). At any rate, HB's result can be formulated as: if the twin prime conjecture is false, then there is no exceptional zero (with an appropriate constant in the definition of the exceptional zero). In other words, there is $c>0$ such that if there exist a quadratic $χ$ modulo q and a real zero $\sigma>1−c/\log q$ of $L(s,\chi)$, then the twin prime conjecture is true. I continue in next comment. –  GH from MO May 19 '13 at 23:00
    
Also, I think your opening line "What is known to follow from the existence of Siegel zeros?" does not reflect what you want (in the light of your comments). You really want: What is known to follow from both the existence and the absence of Siegel zeros. That is, what can we prove unconditionally by using the notion of Siegel zeros on the way. –  GH from MO May 19 '13 at 23:01
    
@GH: The constant $c$ in your statement above (essentially Theorem 2 of HB's paper) is not computable however and it depends on whether the infinite sequence of zeros HB used in his Theorem 1 exists (if it doesn't, we just put $c$ small enough that no zero as specified exists and the implication is trivially true). [HB has a warning about this right after stating Theorem 2.] For my question to make proper sense you should assume that $c$ is given once and for all (cf. my first comment to Tao's answer below), in that case there is no trouble with the notion of a single Siegel zero. –  Kálmán Kőszegi May 20 '13 at 0:10
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4 Answers

With the usual definition of a Siegel zero (involving an unspecified constant $C_\varepsilon$ for each $\varepsilon>0$), it is not easy to talk about a "single" Siegel zero unless one decides to fix exactly how $C_\varepsilon$ is to depend on $\varepsilon$.

On the other hand, the classical proof of the prime number theorem also shows that $L(\sigma+it,\chi)$ has no zeroes in the region $\sigma \geq 1-\frac{c}{\log q(|t|+1)}$ for some effective (and very explicit) $c>0$, with at most one exception. This gives an effective prime number theorem in arithmetic progressions

$$ \psi(x; a,q) = \frac{x}{\phi(q)} - \frac{\chi(a)}{\phi(q)} \frac{x^\beta}{\beta} + O( x \exp(-b \sqrt{\log x}))$$

for an absolute and effective constant $b>0$, where $\beta$ is the exceptional zero (if it exists) of the exceptional quadratic character $\chi$. (If there is no exceptional zero, the second term on the right-hand side is simply deleted.) This formula can then be used as a partial but effective substitute for the Siegel-Walfisz theorem for all sorts of number-theoretic applications, e.g. this formula (or something very close to it) is used in all the known effective unconditional proofs of Vinogradov's three primes theorem. In many cases the results are actually easier to prove if the exceptional zero is present. Iwaniec's ICM survey at http://www.icm2006.org/proceedings/Vol_I/16.pdf discusses these issues in more detail.

ADDED LATER: Another interesting phenomenon, first observed by Montgomery and Weinberger, is that the existence of a single Siegel zero $L(\sigma,\chi)=0$ forces many other L-functions $L(s,\psi)$ to have most of their zeroes (at a certain height) arranged on the critical line and to lie close to an arithmetic progression (this type of behaviour is occasionally referred to as the "Alternative Hypothesis", being the extreme opposite to the more commonly believed "GUE hypothesis" but which thus far has proven impossible to completely exclude). Roughly speaking, the reason for this is that if $L(\sigma,\chi)=0$ for some $\sigma$ close to $1$, then the residue of $\zeta(s) L(s,\chi)$ is unexpectedly small at $1$, making the Dirichlet convolution $1*\chi$ much sparser than expected. For any other Dirichlet character $\psi$, $\psi*\chi\psi$ is pointwise dominated by $1*\chi$ and is similarly sparse. This means that the function $L(s,\psi) L(s,\psi\chi)$ is very well behaved for typical $\psi$ and for $s$ near the critical line; indeed, it is dominated by the initial segment of the Dirichlet series $\sum_n \frac{\psi*\psi\chi(n)}{n^s}$ (which is very smooth in $s$), plus the complementary term coming from the functional equation (or equivalently, from Poisson summation), which oscillates at a precise frequency depending on the height of $s$ and the conductor of $\psi$ and $\psi \chi$. The interaction between these two terms is what places the zeroes of $L(s,\psi) L(s,\psi\chi)$, and hence of $L(s,\psi)$, near an arithmetic progression.

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Is the usual definition $\varepsilon$-dependent like that? At p. 293 in Iwaniec's survey he seems to be providing a definition of "exceptional zero" as a zero $\beta > 1 - \frac{c}{log(D)}$, $c$ being "some fixed sufficiently small positive constant". (My understanding was that $c$ might as well be given some numerical value and was left unspecified for convenience only.) –  Kálmán Kőszegi Mar 23 '13 at 0:07
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The formula above (as well as Ralph's class number formula below) is of course a non-trivial consequence of the existence of $\beta$, but maybe not a truly "good" use of Siegel zeros since $\beta$ itself is turning up in the formula we prove. Since $\beta$ is (we all think?) a fiction this doesn't actually provide us with any true mathematical results we couldn't derive without assuming the existence of $\beta$, even though the formula might be useful for obtaining unconditional results. H-B's twin prime result would be a nice example of what I want, but requires an infinity of zeros... –  Kálmán Kőszegi Mar 23 '13 at 0:26
    
I didn't know this nice survey by Iwaniec. Thanks for the link! –  Joël Mar 23 '13 at 5:55
    
The main application that Montgomery & Weinberger give [the paper is available at: matwbn.icm.edu.pl/ksiazki/aa/aa24/aa2457.pdf] is the determination of all $D$ with $h(D) = 2$, this is just in line with the original Deuring-Heilbronn result though: a known consequence of GRH is shown to follow in the presence of Siegel zeros and we get an unconditional result. I would be interested in the best known bounds for the phenomenon though: assuming a Siegel zero to modulus $q$, how far up the strip can we guarantee that zeros of other L-functions lie on the critical line? –  Kálmán Kőszegi May 20 '13 at 0:39
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There is an asymptotic formula for the relative class number of $p$th-cyclotomic fields where a Siegel zero $\beta$ occurs: $$ h^-(p)=\frac{p+3}{4}\log p-\frac{p}{2}\log 2\pi + \log (1-\beta) +O(\log^2_2 p)$$ If no Siegel zero exists or if $p\equiv 1(4)$, the term $\log (1-\beta)$ disappears in the formula.

This is Theorem 1 from Puchta: On the class number of $p$-th cyclotomic field. Arch. Math. 74 (2000), 266-268. But note the misprint of the sign in the theorem: $\frac{p-3}{4}$ should be $\frac{p+3}{4}$.

Compare also with a classical formula of Masley, Montgomery (Cyclotomic fields with unique factorization. J. Reine Angew. Math. , 286/287 (1976), 248–256. Theorem 1): $$h^-(p)=\frac{p+3}{4}\log p-\frac{p}{2}\log 2\pi+O(\log p)$$

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Is this really a formula for $h^-(p)$, not $\log h^-(p)$? –  Noam D. Elkies May 19 '13 at 19:01
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I just ran across this paper. I don't know if it is a "good" use or not. It is in the spirit of Heath-Brown I presume.

Lithuanian Mathematical Journal January 2010, Volume 50, Issue 1, pp 43-53

L. Germán, I. Kátai, On multiplicative functions on consecutive integers

http://link.springer.com/article/10.1007%2Fs10986-010-9070-8

From the abstract:

Concerning the Liouville function $\lambda$, we find an upper estimate for ${1\over x}|\sum_{n\le x}\lambda(n)\lambda(n+1)|$ under the unproved hypothesis that $L(s,\chi)$ have Siegel zeros for an infinite sequence of L-functions.

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Siegel zeros are motivated by a possible problem with the uniformity of the distribution of primes in progression. As such if there are only finitely many Siegel zeros then the uniformity problem would not arise, and we would say that no Siegel zeros exist!

One of the most striking consequence of the existence of Siegel zeros that I know of, is Heath-Brown's proof that Siegel zeros imply twin prime. It will be interesting to watch and see if this result will be used in the eventual proof of the twin prime conjecture (now that we have bounded gaps between primes).

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Getting the twin prime conjecture from some niceness property of zeros of L-functions (even the full GRH) seems out of reach at present, so in terms of recent results it seems that Helfgott's proof of weak Goldbach is a more promising place to look for applications of anomalous zeros; since WGC is known to follow on the GRH we are free to assume a counterexample $\beta$ for a nonconditional proof. From what I gather from skimming the articles this isn't the strategy he adopted though, only a numerical verification of GRH for many values. –  Kálmán Kőszegi May 19 '13 at 21:13
    
Dear Kalman, As you might know Yitang Zhang has recently shown that there exists infinitely many primes with gaps less than 70 million, for the proof see here: annals.math.princeton.edu/articles/7954 . Many mathematicians have started to think about reducing the 70 million down to a smaller number and eventually to two. In the final attack on the twin prime conjecture, we can assume that Siegel zeros do not exist because of Heath-Brown's result. –  kiskis May 23 '13 at 8:05
    
Another comment on the Goldbach conjecture: Vinogradov's first proof of the ternary Goldbach relied on the Siegel-Walfisz theorem, which in turn depends on Siegel's bound for the Siegel zero. As a result the constant C, such that all odd n > C are a sum of three primes was ineffective. In order to make the constant effective you have to use the weaker, but effective analogue of Siegel-Walfish. This produces an enormous constant. In order to bring the constant down you have to play games with the Siegel zero. –  kiskis May 23 '13 at 8:19
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