2015/09/23 by Adam J. Harper, Harper, Adam J., Youness Lamzouri +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1509.07188
openalex publication_date 2015/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the race between prime numbers in many residue classes modulo\nq, assuming the standard conjectures GRH and LI.\n Among our results we exhibit, for the first time, prime races modulo q with\nn competitor classes where the biases do not dissolve when n, q\→ \∞.\nWe also study the leaders in the prime number race, obtaining asymptotic\nformulae for logarithmic densities when the number of competitors can be as\nlarge as a power of q, whereas previous methods could only allow a power of\n\log q.\n The proofs use harmonic analysis related to the Hardy--Littlewood circle\nmethod to control the average size of correlations in prime number races. They\nalso use various probabilistic tools, including an exchangeable pairs version\nof Stein's method, normal comparison tools, and conditioning arguments. In the\nprocess we derive some general results about orderings of weakly correlated\nrandom variables, which may be of independent interest.\n