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In a surprising recent work, Lemke Oliver and Soundararajan noticed howexperimental data exhibits erratic distributions for consecutive pairs ofprimes in arithmetic progressions, and proposed a heuristic model based on theHardy--Littlewood conjectures containing a large secondary term, which fits thedata very well. In this paper, we study consecutive pairs of sums of squares inarithmetic progressions, and develop a similar heuristic model based on theHardy--Littlewood conjecture for sums of squares, which also explain the biasesin the experimental data. In the process, we prove several results related toaverages of the Hardy--Littlewood constant in the context of sums of twosquares.

Lemke Oliver and Soundararajan bias for consecutive sums of two squares