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 $3\mathrm{mod}4$ up to $x$ than of the form $1\mathrm{mod}4$. 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 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 $1\mathrm{mod}3$ is greater than those that are $2\mathrm{mod}3$ 100% of the time in natural density sense.

arXiv commentAdded mention of work of M. Radziejewski, fixed typos. Accepted version. 25 pages, 3 figures, 2 tables

References in corpus (10)

• Real zeros of quadratic Dirichlet L-functions
• Chebyshev's bias for analytic L-functions
• Highly biased prime number races
• Big biases amongst products of two primes
• Chebyshev's bias for products of irreducible polynomials
• Chebyshev's bias in dihedral and generalized quaternion Galois groups
• Chebyshev's bias for products of $k$ primes
• Lemke Oliver and Soundararajan bias for consecutive sums of two squares
• Character sums over products of prime polynomials
• The variance of the number of sums of two squares in ${\mathbb{F}}_q[T]$ in short intervals

Cited by in corpus (2)

• An annotated bibliography for comparative prime number theory
• Positive density for consecutive runs of sums of two squares