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Sums of two squares are strongly biased towards quadratic residues

Ofir Gorodetsky

Vol. 17 (2023), No. 3, 775–804
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 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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