Bilinear forms in the Fourier coefficients of half-integral weight cusp forms and sums over primes
type: Item
instance of: scholarly article
title: Bilinear forms in the Fourier coefficients of half-integral weight cusp forms and sums over primes
zbMATH Open document ID: 0671.10021
DOI: 10.1007/BF01453602
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164664
review text: Let \(\hat f_ n\) be the n-th Fourier coefficient of a holomorphic cusp form of weight \(k=1/2+\ell \geq 5/2\) for \(\Gamma_ 0(N)\), \(N\equiv 0(mod 4)\) and \(f_ n=\hat f_ nn^{-(k-1)/2}.\) For arbitrary complex numbers \(a_ m\) supported on square-free integers it is shown that \[ \sum_{n\leq X}\quad | \sum_{m\leq Y}a_ mf_{mn}|^ 2\quad \ll \quad (X+X^{1/2}Y^ 2)(XY)^{\epsilon} \| a\|^ 2 \] for any X,Y\(\geq 1\) and \(\epsilon >0\), the implied constant depending only on f and \(\epsilon\). Stronger estimates for bilinear forms with special coefficients are also given. These are combined to yield the following estimate for the sum over primes \[ \sum_{p<X}f_ p\quad \ll \quad X^{155/156+\epsilon}, \] where again the implied constant depends only on f and \(\epsilon\). Proofs are based on an exponential sums alternative to the Rankin-Selberg method.
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11L03
zbMATH DE Number: 4097478
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: holomorphic cusp form
zbMATH Keywords: estimates for bilinear forms
zbMATH Keywords: sum over primes
zbMATH Keywords: exponential sums
author: W. Duke
author: H. Iwaniec
MaRDI profile type: Publication
cites work: Q4200285
cites work: On the Mean Value of L(1/2, χ ) FW Real Characters
cites work: Q4154638
cites work: On modular forms of half integral weight
cites work: Mean Value Theorems in Prime Number Theory
OpenAlex ID: W2040637330
description: scientific article; zbMATH DE number 4097478