An asymptotic formula for a trigonometric sum of vinogradov

We obtain a representation formula for the trigonometric sum f(m, n) := Sig ma (m-1)(a=1) (sin(pia/m))/(\ sin(pi an/m)\) and deduce from it the upper b ound f(m, n) < (4/pi (2)) m log m + (4/pi (2))(y-log(pi /2)+2C(G)) m+O(m/ro ot log m), where C-G is the supremum of the function G(t) := Sigma (alpha)( k=1) log \2 sin pi kt \/(4k(2)-1), over the set of irrationals. The coeffic ients on both the main term and the second term are shown to be best possib le. This improves earlier bounds for f(m, n). It is conjectured that C-g =G (root2)approximate to 0.236. We also obtain the following asymptotic formul a: If alpha is a real algebraic integer of degree 2 with 0 < alpha < 1, the n for any rational approximation n/m of alpha with 0 < n < m we have f (m, n) = (4/pi (2)) m log m+(4 pi (2))(y- log(pi z2)+2G(alpha)) m+ O-alpha (\ a lpha-(n)(m/)\ (1/2) m log m). (C) 2001 Academic Press.