On a Trigonometric Inequality of Vinogradov

AbstractThe sum ƒ(m, n) = ∑m − 1a = 1 (|sin(xan/m)|/sin(xa/m)) arises in bounding incomplete exponential sums. In this article we show that for positive integers m, n with m > 1, ƒ(m, n) < (4m/π2)(log m + γ + 18 − log(π/2)) + (2/π)(2 − 1/π), where γ is Euler′s constant. This improves earlier bounds for ƒ(m, n)