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 "exceptional 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?
Best Answer
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.