The variance of primes in quadratic progressions: manuscript, code and data
Year: 2026
Type: preprint
Abstract: Manuscript (PDF and LaTeX source), code (Python) and data of the project 'The variance of primes in quadratic progressions'. Main results: an unconditional asymptotic formula with a secondary term for the second moment of the pair singular series over the quadratic shifts f_c(k') - f_c(k), an explicit window term for a single window, and, under the Hardy-Littlewood prime pair conjecture with a power saving, a Montgomery-Soundararajan type law of variance for the number of primes among k^2 + bk + c (b in {0,1}) in short windows, averaged over the position of the window; for Y >= x^{2/3+eps} the conjecture is not needed. Numerical experiments (two runs with c up to 2*10^5, with error analysis) are included; all result files carry SHA-256 hashes. The proofs, the computations and the text were prepared with substantial use of an AI assistant (Claude, Anthropic) and checked by independent AI reviews; at the time of this deposit they have not been verified by an independent human expert. The authors take full responsibility for the contents. Archive built by rekopis/zbuduj_pakiet.py; the exact source commit and SHA-256 hashes of every file are listed in MANIFEST.txt inside this archive and are also encoded in the archive's file name.",
Authors: Mrugała, Mateusz, Szczypkowski, Jakub
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225639380
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