Journal article
Bounds toward Hypothesis S for cusp forms
Journal of number theory, Vol.236, pp.128-143
07/2022
DOI: 10.1016/j.jnt.2021.07.012
Abstract
Iwaniec, Luo, and Sarnak proposed Hypothesis S and its generalization which predicts non-trivial bounds for a smooth sum of the product of an arithmetic sequence {an} and a fractional exponential function. When an is the Fourier coefficient λf(n) of a fixed holomorphic cusp form f, however, a resonance phenomenon prohibits any improvement of the bound beyond a barrier. It is believed that this resonance barrier could be overcome when the weight k of f tends to infinity. The present paper is a first step toward this goal by proving non-trivial bounds for this sum when k and the summation length X both tend to infinity. No such non-trivial bounds are previously known if the form f is allowed to move. Similar bounds are also proved for linear phases and for Maass forms. The main technology is improved large sieve inequalities over a short interval.
Details
- Title: Subtitle
- Bounds toward Hypothesis S for cusp forms
- Creators
- Yangbo Ye - University of Iowa
- Resource Type
- Journal article
- Publication Details
- Journal of number theory, Vol.236, pp.128-143
- DOI
- 10.1016/j.jnt.2021.07.012
- ISSN
- 0022-314X
- eISSN
- 1096-1658
- Publisher
- Elsevier Inc
- Language
- English
- Date published
- 07/2022
- Academic Unit
- Mathematics
- Record Identifier
- 9984241158402771
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