A theorem of Bombieri–Vinogradov type with few exceptional moduli
Volume 195 / 2020
Acta Arithmetica 195 (2020), 313-325
MSC: Primary 11N13.
DOI: 10.4064/aa190527-7-1
Published online: 8 May 2020
Abstract
Let $1 \le Q \le x^{9/40}$ and let $\mathcal S$ be a set of pairwise relatively prime integers in $[Q,2Q)$. The prime number theorem for arithmetic progressions in the form \[\max _{y\le x}\max _{\substack {a\\ (a,q)=1}} \bigg |\sum _{\substack { n \equiv a\pmod q\\ n \le y}} \Lambda (n) - \frac x{\phi (q)}\bigg | \lt \frac x{\phi (q)(\log x)^A}\] holds for all $q$ in $\mathcal S$ with $O((\log x)^{34+A})$ exceptions.
