On a trigonometric sum of Vinogradov
SourceJournal of Number Theory
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The trigonometric sum f(m, n) = ∑ k=1 m-1 sin(πkn/m) /sin(πk/m) (1 < m ∈ N, n ∈ N) has several applications in number theory. We prove that the mean value inequalities c1m(log m + γ - logπ/2) ≤ 1/m ∑ n=1 f(m, n) < c2m(log m + γ - logπ/2) (m = 2, 3,...) hold with the best possible constant factors c1 = 1/4[γ + log(4/π)] = 0.30533... and c 2 = 4/π2 = 0.40528... . This result refines and complements inequalities due to Cochrane, Peral, and Yu. © 2003 Elsevier Inc. All rights reserved.