An asymptotic formula for a trigonometric sum of Vinogradov
The authors give a representation formula and an asymptotic formula for the trigonometric sum \[ f(m,n):= \sum_{k=1}^{m-1} \frac{|\sin ( \pi k n /m)|}{\sin ( \pi k /m)}. \] As an application they deduce from the representation formula the upper bound \[ f(m,n) < (4/ \pi^2) m \log m + (4/ \pi^2) ( \gamma - \log ( \pi /2) + 2 C_G) m + O(m/ \sqrt{\log m}), \] where \(C_G\) is the supremum of the function \(G(t) := \sum_{k=1}^{\infty} \log |2 \sin \pi k t|/(4k^2-1)\), over the set of irrationals. The coefficients on both the main term and the second term are shown to be best possible. This improves earlier bounds for \(f(m,n)\).
scientific article; zbMATH DE number 1787783