Multiplicative Functions on Arithmetic Progressions

Abstract

Abstract.Let / be a multiplicative arithmetic function satisfying \f\ < 1 , let x > 10 and 2 < Q < x1/3 . It is shown that, with suitable integers q\>2 and qi > 2 , the estimate n=a mod q {n,q)=\ holds uniformly for (a,q) = 1 and all moduli q < Q that are not multiples of Q\ or q2 .

Multiplicative functions in arithmetic progressions

“…Elliott [4,5] showed that this is ≪ x(log log x/ log x) 1 8 for all q except possibly for multiples of a certain exceptional modulus r: note that this is non-trivial in the range q ≤ (log x) 1 8 +o(1) . Hildebrand [14] showed that (1.7) is ≪ x/(q √ log A) for all q ≤ Q except possibly for multiples of one of at most two exceptional moduli r and r ′ . Our result improves both of these, by understanding the asymptotics in all cases, at worst in terms of the exceptional moduli.…”