Sums of two squares are strongly biased towards quadratic residues

TL;DR

This paper demonstrates a strong bias in the distribution of sums of two squares across arithmetic progressions, showing they are more often quadratic residues, with results supported by a Chowla-type conjecture and under GRH.

Contribution

It reveals a significantly stronger bias in sums of two squares in arithmetic progressions than in primes, requiring only a Chowla-type conjecture instead of zero independence assumptions.

Findings

Sums of two squares are more often quadratic residues in arithmetic progressions.

Under GRH, the bias is observed in 100% of cases in natural density.

The bias is stronger than that observed in prime distributions.

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 on the 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 nonvanishing 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 $2 \bmod 3$ 100% of the time in natural density sense.