Bilinear forms with Kloosterman fractions
type: Item
instance of: scholarly article
title: Bilinear forms with Kloosterman fractions
zbMATH Open document ID: 0873.11050
author: W. Duke
author: J. B. Friedlander
author: H. Iwaniec
published in: Inventiones Mathematicae
publication date: 1997-10-30T00:00:00Z
review text: The Kloosterman sum with character \(\chi \pmod c\) is \[ S_\chi (a,b;c) = \sum_{y \pmod c} \overline \chi(y)e \left({a \overline y+by \over c} \right) \] where \(e(t)= e^{2\pi it}\) and \(\overline yy \equiv 1 \pmod c\). If \(\chi\) is the principal character this is the classical Kloosterman sum and if \(\chi(y) =({y \over c})\) is the Jacobi symbol then this is the Salié sum. In applications one usually needs estimates for incomplete sums where the variable runs over a segment of an arithmetic progression. A standard method of bounding such a sum is to express it in terms of the complete sums \(S_\chi (a,b;c)\) by a Fourier technique and then apply Weil's bound for the latter. When the segment is too small this method gives nothing. Fortunately, for many applications it suffices to bound these sums on average over the modulus \(c\). In this paper the authors consider general bilinear forms of the type \[ {\mathcal B} (M,N)= {\underset {(m,n)=1} {\sum\sum}} \alpha_m \beta_ne \left(a {\overline m \over n} \right), \] where \(a\) is a possibly large positive integer and \(\alpha_m\), \(\beta_n\) are arbitrary complex numbers for \(M <m \leq 2M\), \(N<n \leq 2N\), respectively and \(\overline mm \equiv 1 \pmod n\). A trivial bound for this is \(|{\mathcal B} (M,N) |\leq 2 |\alpha ||\beta|(MN)^{1 \over 2}\) where \(|\cdot |\) denotes the \(\ell_2\)-norm and the authors' goal is to improve this. Their first result is: Theorem 1. For any positive integer \(a\) we have \[ {\mathcal B} (M,N) \ll |\alpha ||\beta |\left \{(M+N)^{1\over 2} +\left(1 +{a \over MN} \right)^{1 \over 2} \min (M,N) \right\} (MN)^\varepsilon \] where the implied constant depends only on \(\varepsilon\). Theorem 1 provides nontrivial bounds except in the case that one of the ranges \(M,N\) is much larger than the other and in the case when they are nearly equal. In the former case one has essentially a single variable and one cannot in general obtain nontrivial bounds since the unknown coefficients …
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11E39
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 1026478
zbMATH Keywords: automorphic forms
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: Artin \(L\)-functions of degree two
zbMATH Keywords: mean value theorems for Dirichlet \(L\)-functions
zbMATH Keywords: Kloosterman sum
zbMATH Keywords: bilinear forms
zbMATH Keywords: Salié sum
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: theta multiplier
zbMATH Keywords: Linnik problem
zbMATH Keywords: convexity bounds
zbMATH Keywords: Dirichlet polynomials
zbMATH Keywords: divisor functions
zbMATH Keywords: primes in arithmetic progressions
MaRDI profile type: Publication
full work available at URL: s002220050135
OpenAlex ID: W2075483325
DOI: 10.1007/S002220050135
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description: scientific article; zbMATH DE number 1026478