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A note on multiplicative functions on progressions to large moduli

Part of: Multiplicative number theory

Published online by Cambridge University Press:  04 December 2017

Ben Green
Ben Green*
Affiliation:
Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK (benjosephgreen@gmail.com)
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Abstract

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Let f : ℕ → ℂ be a bounded multiplicative function. Let a be a fixed non-zero integer (say a = 1). Then f is well distributed on the progression n ≡ a (mod q) ⊂ {1,…, X}, for almost all primes q ∈ [Q, 2Q], for Q as large as X1/2+1/78−o(1).

Keywords

MöbiusLiouvillemultiplicative functionbilinear form

MSC classification

Secondary: 11N64: Other results on the distribution of values or the characterization of arithmetic functions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 1 , February 2018 , pp. 63 - 77
Copyright
Copyright © Royal Society of Edinburgh 2018