Sums of two squares are strongly biased towards quadratic residues

Gorodetsky, Ofir

doi:10.2140/ant.2023.17.775

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Sums of two squares are strongly biased towards quadratic residues

Ofir Gorodetsky

Vol. 17 (2023), No. 3, 775–804

DOI: 10.2140/ant.2023.17.775

Abstract

Chebyshev famously observed empirically that more often than not, there are more primes of the form $3 mod 4$ up to $x$ than 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 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 $\frac{1}{2}$ . To illustrate, we have under GRH that the number of sums of two squares up to $x$ that are $1 mod 3$ is greater than those that are $2 mod 3$ 100% of the time in natural density sense.

Keywords

Chebyshev's bias, sums of two squares, omega function, prime divisor function

Mathematical Subject Classification

Primary: 11N37

Secondary: 11M06

Milestones

Received: 30 November 2021

Revised: 1 May 2022

Accepted: 21 June 2022

Published: 12 April 2023

Authors

Ofir Gorodetsky
Mathematical Institute University of Oxford Oxford United Kingdom

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Vol. 17, No. 3, 2023