2025/04/28 by Elena Fuchs, Fuchs, Elena, Catherine Hsu +7
Mathematics · #11D09 #11E12 #11N32 #11N36 #52C26 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2504.20289
openalex publication_date 2025/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove lower bounds of the form ≫ N/(log N)3/2 for the number of primes up to N primitively represented by a shifted positive definite integral binary quadratic form, and under the additional condition that primes are from an arithmetic progression. This extends the sieve methods of Iwaniec, who showed such lower bounds without the primitivity and congruence conditions. Imposing primitivity adds some subtleties to the local criteria for representation of a shifted prime: for example, some shifted quadratic forms of discriminant 5 \pmod8 do not primitively represent infinitely many primes. We also provide a careful list of the local conditions under which a genus of an integral binary quadratic form represents an integer, verified by computer, and correcting some minor errors in previous statements. The motivation for this work is as a tool for the study of prime components in Apollonian circle packings [FFH+24]