On a trigonometric inequality of vinogradov

Abstract

The sum $\text{f(m, n)=}\text{∑}{}_{\mathrm{a=1}}{}^{\mathrm{m-1}}\text{(}\text{|}\text{sin}\text{πan}\text{m}\text{|}\text{|}\text{sin}\text{πa}\text{m}\text{|)}$ arises in bounding incomplete exponential sums. In this article we show that for positive integers m, n with m>1, $\text{f(m, n)<(}\text{4}\text{π}{}^2\text{) m}\text{log}\text{m+0.38m+0.608+0.116}\text{d}{}^2\text{m}$, where d=(m, n). This improves earlier bounds for f(m, n). The constant $\text{4}\text{π}{}^2$ in the main term is shown to be best possible.