2021/07/09 by Stephan Baier, Baier, Stephan, Sudhir Pujahari +1
Mathematics · Arts and Humanities · #Analytic Number Theory Research #Historical Studies and Socio-cultural Analysis #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2107.04348
In 2020, Roger Baker \citeBak proved a result on the exceptional set of moduli in the prime number theorem for arithmetic progressions of the following kind. Let S be a set of pairwise coprime moduli q≤ x9/40. Then the primes l≤ x distribute as expected in arithmetic progressions mod q, except for a subset of S whose cardinality is bounded by a power of log x. We use a p-adic variant Harman's sieve to extend Baker's range to q≤ x1/4-ε if S is restricted to prime powers pN, where p≤ (log x)C for some fixed but arbitrary C>0. For large enough C, we thus get an almost all result. Previously, an asymptotic estimate for π(x;pN,a) of the expected kind, with p being an odd prime, was established in the wider range pN≤ x3/8-ε by Barban, Linnik and Chudakov \citeBLC. Gallagher \citeGal extended this range to pN≤ x2/5-ε and Huxley \citeHux2 improved Gallagher's exponent to 5/12. A lower bound of the correct order of magnitude was recently established by Banks and Shparlinski \citeBaS for the even wider range pN≤ x0.4736. However, all these results hold for \it fixed primes p, and the O-constants in the relevant estimates depend on p. Therefore, they do not contain our result. In a part of our article, we describe how our method relates to these results.