Primes in Arithmetic Progressions to Large Moduli. III

Abstract

q<x'l' log-II x (q .a)=1-A In(x; q ,a) -n(x)!</J(q)1 «:A x log x.This deals with essentially the same range of q as can be treated conditionally under GRH for individual progressions.For significantly larger q, the asymptotic formula is not known to hold, even in the above averaged sense, but there do follow by sieve methods upper bounds of the expected order of magnitude under the very mild restriction q < x 1-£ .Known as Brun-Titchmarsh theorems, these bounds hold both for the individual

Locations

• Journal of the American Mathematical Society - View - PDF