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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.

11L07 Estimates on exponential sums
42A05 Trigonometric polynomials, inequalities, extremal problems
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[1] Chalk, J.H.H., The number of solutions of congruences in incomplete residue systems, Canad. J. math., 15, 291-296, (1963) · Zbl 0112.27005
[2] Chalk, J.H.H.; Williams, K.S., The distribution of solutions of congruences, Mathematika, 12, 176-192, (1965) · Zbl 0158.29603
[3] Gradshteyn, T.S.; Ryzhik, I.M., ()
[4] Lidl, R.; Niederreiter, H., ()
[5] Mordell, L.J., The number of solutions in incomplete residue sets of quadratic congruences, Arch. math., 8, 153-157, (1957) · Zbl 0079.06306
[6] Mordell, L.J., Incomplete exponential sums and incomplete residue systems for congruences, Czech. math. J., 14, 235-242, (1964) · Zbl 0135.10502
[7] Polya, G.; Szego, G., ()
[8] Serre, J., Majorations de sommes exponentielles, Asterisque, 41-42, 111-126, (1977) · Zbl 0406.14014
[9] Smith, R.A., The distribution of rational points on hypersurfaces defined over a finite field, Mathematika, 17, 328-332, (1970) · Zbl 0228.14015
[10] Spackman, K., On the number and distribution of simultaneous solutions to diagonal congruences, Canad. J. math., 33, No. 2, 421-436, (1981) · Zbl 0411.12015
[11] Vinogradov, I.M., ()
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