Granville, A;
Harper, AJ;
Soundararajan, K;
(2019)
A new proof of Halasz's theorem, and its consequences.
Compositio Mathematica
, 155
(1)
pp. 126-163.
10.1112/S0010437X18007522.
Preview |
Text
Granville HalaszActualPaperRevision.pdf - Accepted Version Download (491kB) | Preview |
Abstract
Abstract. Hal´asz’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 |f(n)| ≤ 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).
| Type: | Article |
|---|---|
| Title: | A new proof of Halasz's theorem, and its consequences |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1112/S0010437X18007522 |
| Publisher version: | https://doi.org/10.1112/S0010437X18007522 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | multiplicative functions, Halász’s theorem, Hoheisel’s theorem, Linnik’s theorem, Siegel zeroes |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/1568743 |
Archive Staff Only
 |
View Item |
