Recently Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the Hardy–Littlewood conjectures containing a large secondary term, which fits the data very well. We will discuss the analogous question for consecutive pairs of sums of squares in arithmetic progressions, a bias also appears in the experimental data and we develop a similar heuristic model based on the Hardy–Littlewood conjecture for sums of squares to explain it.
This is joint work with Chantal David, Jungbae Nam and Jeremy Schlitt.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/246