An Asymptotic Formula for a Trigonometric Sum of Vinogradov Original Research Article

We obtain a representation formula for the trigonometric sum f(m, n)colon, equals ∑m−1a=1 image and deduce from it the upper bound f(m, n)<(4/π2) m log m+ (4/π2)(γ−log(π/2)+2CG) m+O(m/image), where CG is the supremum of the function G(t)colon, equals∑∞k=1 log 2 sin πkt/(4k2−1), 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). It is conjectured that CG=G(image)≈ 0.236. We also obtain the following asymptotic formula: If α is a real algebraic integer of degree 2 with 0<α<1, then for any rational approximation n/m of α with 0