An asymptotic formula for a trigonometric sum of Vinogradov
type: Item
instance of: scholarly article
title: An asymptotic formula for a trigonometric sum of Vinogradov
zbMATH Open document ID: 0998.11043
author: Todd Cochrane
author: Juan Carlos Peral
published in: Journal of Number Theory
publication date: 2002-08-25T00:00:00Z
review text: The authors give a representation formula and an asymptotic formula for the trigonometric sum \[ f(m,n):= \sum_{k=1}^{m-1} \frac{|\sin ( \pi k n /m)|}{\sin ( \pi k /m)}. \] As an application they deduce from the representation formula the upper bound \[ f(m,n) < (4/ \pi^2) m \log m + (4/ \pi^2) ( \gamma - \log ( \pi /2) + 2 C_G) m + O(m/ \sqrt{\log m}), \] where \(C_G\) is the supremum of the function \(G(t) := \sum_{k=1}^{\infty} \log |2 \sin \pi k t|/(4k^2-1)\), over the set of irrationals. The coefficients on both the main term and the second term are shown to be best possible. This improves earlier bounds for \(f(m,n)\).
Mathematics Subject Classification ID: 11L07
Mathematics Subject Classification ID: 42A05
zbMATH DE Number: 1787783
zbMATH Keywords: representation formula
zbMATH Keywords: asymptotic formula
zbMATH Keywords: trigonometric sum
zbMATH Keywords: upper bound
reviewed by: Jürgen Hinz
MaRDI profile type: Publication
full work available at URL: jnth.2001.2679
OpenAlex ID: W1988899221
cites work: On a trigonometric inequality of Vinogradov
cites work: On a sum of Vinogradov
cites work: Q5666618
cites work: Q5830713
DOI: 10.1006/JNTH.2001.2679
description: scientific article; zbMATH DE number 1787783