There are few results that I am aware of where one can prove something stronger by assuming the existence of Siegel zeros than by assuming the GRH. For example Heath-Brown proved the existence of Siegel zeros imply the twin prime conjecture while it is unknown under GRH. Also there is: Let $P(a,q)$ be the least prime $\equiv a \pmod q$. Then assuming GRH we have $P(a,q) \ll q^L$ where $L < 2 + \varepsilon$, but assuming the existence of the Sigel zeros we have $L < 2$. (This I just took from Good uses of Siegel zeros?)
I was interested in this phenomenon. I would greatly appreciate if anyone could explain why or give some ideas on why this is the case?
Also this is a question with subjective answer, but when there is a result of this type (where by assuming Siegel zeros one gets even stronger results than assuming GRH), do we generally expect it to be the truth? Thank you.
Any comments are appreciated.
Best Answer
Roughly speaking, GRH asserts that the Möbius function $\mu$ is "orthogonal" to all Dirichlet characters $\chi$, in the sense that correlations such as $\sum_{n \leq x} \mu(n) \overline{\chi(n)}$ are very small. This is the expected behaviour of the Möbius function, and through various standard analytic number theory manipulations one can also use GRH to control correlations between the von Mangoldt function $\Lambda(n)$ (which basically encodes primes) and various other functions, e.g. linear phases $e(\alpha n)$. On the other hand, GRH struggles to control self-correlations such as $\sum_{n \leq x} \mu(n) \mu(n+2)$ or $\sum_{n \leq x} \Lambda(n) \Lambda(n+2)$ (the latter being used to count twin primes). For instance, in the function field case GRH is a known theorem, but we are still unable to obtain an asymptotic (or even a lower bound of the right order of magnitude) for twin primes, though it is possible to establish the infinitude of twin primes in this case by more algebraic means.
In contrast, the existence of a Siegel zero means that there is a quadratic Dirichlet character $\chi$ with which $\mu$ has very high correlation (this can be quantified precisely using the "pretentious" approach to analytic number theory developed by Granville and Soundararajan). This would be very unusual behavior for $\mu$, and as such is actually a rather powerful piece of information. As a crude first approximation, a Siegel zero allows one to replace $\mu$ with $\chi$ with acceptable error (in practice one has to be more careful at small primes, though, for instance by imposing a suitable preliminary sieve). Thus for instance one could hope to approximate $\sum_{n \leq x} \mu(n) \mu(n+2)$ by something resembling $\sum_{n \leq x} \chi(n) \chi(n+2)$, which is relatively easy to bound nontrivially. Very roughly speaking, Heath-Brown's arguments proceed by similarly replacing the von Mangoldt function $\Lambda(n) = \sum_{d|n} \mu(d) \log \frac{n}{d}$ with the variant $f(n) := \sum_{d|n} \chi(d) \log \frac{n}{d}$, which is roughly of the same order of complexity as the divisor function; in particular sums such as $\sum_{n \leq x} f(n) f(n+2)$ are tractable enough by known methods (e.g. Kloosterman sum bounds) that they are a useful approximant to the otherwise intractable $\sum_{n \leq x} \Lambda(n) \Lambda(n+2)$.