Journal
COMPOSITIO MATHEMATICA
Volume 155, Issue 1, Pages 126-163Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1112/S0010437X18007522
Keywords
multiplicative functions; Halasz's theorem; Hoheisel's theorem; Linnik's theorem; Siegel zeroes
Categories
Funding
- European Research Council [670239]
- NSERC Canada under the CRC program
- Centre de recherches matheematiques in Montreeal
- Jesus College, Cambridge
- NSF [DMS 1500237, DMS 1440140]
- Simons Foundation
Ask authors/readers for more resources
Halasz's theorem gives an upper bound for the mean value of a multiplicative function f. The bound is sharp for general such f, and, in particular, it implies that a multiplicative function with vertical bar f(n)vertical bar <= 1 has either mean value 0, or is 'close to' n(it) for some fixed t. The proofs in the current literature have certain features that are difficult to motivate and which are not particularly flexible. In this article we supply a different, more flexible, proof, which indicates how one might obtain asymptotics, and can be modified to treat short intervals and arithmetic progressions. We use these results to obtain new, arguably simpler, proofs that there are always primes in short intervals (Hoheisel's theorem), and that there are always primes near to the start of an arithmetic progression (Linnik's theorem).
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available