On a conjecture of Chalk
type: Item
instance of: scholarly article
title: On a conjecture of Chalk
zbMATH Open document ID: 0876.11044
author: Ping Ding
published in: Journal of Number Theory
publication date: 1997-07-24T00:00:00Z
review text: Let \(f \in \mathbb{Z}[x]\) be a polynomial of degree \(g \geq 2\). Consider, for any positive integer \(q\), the complete trigonometric sum \[ S(q,f) = \sum_{k=0}^{q-1}\exp (2 \pi i f(k)/q). \] In 1987 \textit{J. H. H. Chalk} [Mathematika 34, 115-123 (1987; Zbl 0621.10024)] made a conjecture on an upper bound for \(S(q,f)\), when \(q\) is a prime power \(p^n\). In the present paper the author proves this conjecture if \(p\) is relatively small but \(\geq\) 3. When \(p \geq 3\) is relatively large, he gives a counterexample to Chalk's conjecture and derives an alternative upper bound which is best possible. Moreover, for \(p=2\), the author improves previous results.
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 1037297
zbMATH Keywords: conjecture of Chalk
zbMATH Keywords: trigonometric sum
zbMATH Keywords: estimates for exponential sums
Wikidata QID: Q123014038
reviewed by: Jürgen Hinz
MaRDI profile type: Publication
full work available at URL: jnth.1997.2125
OpenAlex ID: W2002760437
cites work: On Hua's estimates for exponential sums
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cites work: Q3856819
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cites work: On Hau's lemma
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DOI: 10.1006/JNTH.1997.2125
description: scientific article; zbMATH DE number 1037297