A sketch of the proof of non-existence of Siegel zeroes

For a given prime , the Legendre symbol is defined as follows: if is a quadratic residue modulo , then ; if is a quadratic non-residue modulo , then ; and if divides , then . Let be an integer that is not a perfect square. (One could require to be a fundamental discriminant if desired, although it turns out to be immaterial for the arguments here.) By combining quadratic reciprocity with the prime number theorem in arithmetic progressions, one can show that the Legendre symbol is asymptotically balanced between and . For instance, one can show that as , where the sum is over primes ; equivalently (by Mertens’ theorem), However, most of the known proofs of this asymptotic are ineffective with regards to the decay of the error. This is due to the infamous Siegel zero problem, arising from Dirichlet -functions associated to with a zero very close to . A well known example comes when , which is a non-residue for all primes ; this is closely tied to Euler’s famous observation that is prime…