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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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##### References:
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