Sums of two squares are strongly biased towards quadratic residues

Journal article

Chebyshev famously observed empirically that more often than not, there are more primes of the form 3mod4 up to x than of the form 1mod4. 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 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 12. To illustrate, we have under GRH that the number of sums of two squares up to x that are 1mod3 is greater than those that are 2mod3 100% of the time in natural density sense.

Authors: Gorodetsky, O

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Mathematical Sciences Publishers
• Journal: Algebra and Number Theory
• Volume: 17
• Issue: 3
• Pages: 775-804
• Publication date: 2023-04-12
• Acceptance date: 2022-06-21
• DOI: 10.2140/ant.2023.17.775
• EISSN: 1944-7833
• ISSN: 1937-0652
• Language: English
• Keywords: sums of two squares; omega function; Chebyshev's bias; prime divisor function; FFR