Successive sums of two squares in arithmetic progressions

Zoom ID: 919 6497 4060
For password please contact Pieter Moree (moree@mpim-bonn.mpg.de)

Joint work with L. Devin, J. Nam and J. Schlitt.
In general, we expect that sets of integers with reasonable multiplicative constraints, as primes and sums of two squares, to be well distributed in arithmetic progressions and small intervals. We study in this talk the finer (and more difficult) question of the distribution of successive sums of two squares in arithmetic progressions modulo q. Probabilistic models predics that each pair of residues (a, b) (mod q) will contain the same number of sums of two squares (asymptotically), but the numerical data exhibit significant fluctuations between the q^{^2} classes (a, b), depending on whether a = b or not.
Inspired by the recent work of Lemke Oliver and Soundararajan, who studied successive primes in arithmetic progressions, we present a model based on the conjectures of HardyLittlewood (for sums of two squares) which predicts an asymptotic explaining the fluctuations between the classes (a, b). The conjecture that we obtain from this model reduces the asymptotic to an average of the Hardy-Littlewood constants, which we compute with enough precision to include (large) secondary terms explaining the bias of the numerical data. Our results also improve previous result of the literature on averages of Hardy-Littlewood constants for sums of two squares.