Chebyshev famously observed empirically that more often than not, there are more primes of the form 3 mod 4 up to x than primes of the form 1 mod 4. This was confirmed theoretically much later by Rubinstein and Sarnak in a logarithmic density sense. Our understanding of this is conditional on the generalized Riemann Hypothesis as well as Linear Independence of the zeros of L-functions.
We investigate similar questions for sums of two squares in arithmetic progressions. We find a significantly stronger bias than in primes, which happens for almost all integers in a natural density sense. Because the bias is more pronounced, we do not need to assume Linear Independence of zeros, only a Chowla-type Conjecture on non-vanishing of L-functions at 1/2.
We'll aim to be self-contained and define all the notions mentioned above during the talk. We shall review the origin of the bias in the case of primes and the work of Rubinstein and Sarnak. We'll explain the main ideas behind the proof of the bias in the sums-of-squares setting.