On Gaussian primes in sparse sets

Journal article

We show that there exists some such that, for any set of integers B with for all , there are infinitely many primes of the form with . We prove a quasi-explicit formula for the number of primes of the form with for any with and , in terms of zeros of Hecke L-functions on . We obtain the expected asymptotic formula for the number of such primes provided that the set B does not have a large subset which consists of multiples of a fixed large integer. In particular, we get an asymptotic formula if B is a sparse subset of primes. For an arbitrary B we obtain a lower bound for the number of primes with a weaker range for , by bounding the contribution from potential exceptional characters.

Authors: Merikoski, J

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Cambridge University Press
• Journal: Compositio Mathematica
• Volume: 161
• Issue: 2
• Pages: 181-243
• Publication date: 2025-06-17
• Acceptance date: 2024-09-24
• DOI: 10.1112/s0010437x24007632
• EISSN: 1570-5846
• ISSN: 0010-437X
• Language: English
• Keywords: Gaussian prime numbers; 11N32; large sieve; 11N36; sieve methods; Siegel zeros; character sums; Hecke L-functions