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On a trigonometric inequality of Vinogradov. (English) Zbl 0629.10030

For positive integers \(m, n\) with \(m>1\) let \[ f(m,n)=\sum_{a=1}^{m-1}| \sin (\pi an/m)| / | \sin (\pi a/m)|. \] This sum arises in bounding incomplete exponential sums. I. M. Vinogradov [Elements of number theory. New York: Dover (1954; Zbl 0057.28201)] showed that \(f(m,n)<m \log m+O(m)\), and this was improved by the reviewer [Math. Comput. 30, 571–597 (1976; Zbl 0342.65002)] to \(f(m,n)<(2/\pi)m \log m+O(m)\). In the present paper it is proved that \(f(m,n)<(4/\pi^2)m \log m+O(m)\) and that the constant \(4/\pi^2\) in the main term is best possible. The constants in all the \(O\)-terms are explicit.

MSC:

11L07 Estimates on exponential sums
42A05 Trigonometric polynomials, inequalities, extremal problems
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