Sums of two squares are strongly biased towards quadratic residues

Abstract

Chebyshev famously observed empirically that more often than not, there are more primes of the form $3 \bmod 4$ up to $x$ than of the form $1 \bmod 4$. 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 \emph{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 $1/2$. To illustrate, we have under GRH that the number of sums of two squares up to $x$ that are $1 \bmod 3$ is greater than those that are $2 \bmod 3$ for all but $o(x)$ integers

Locations

• arXiv (Cornell University) - View - PDF