A mean value estimate for real character sums

A hybrid mean value involving a new Gauss sums and Dedekind sums

‎In this paper‎, ‎we introduce a new sum‎ ‎analogous to Gauss sum‎, ‎then we use the properties of the‎ ‎classical Gauss sums and analytic method to study the hybrid mean‎ ‎value problem involving this new sums and Dedekind sums‎, ‎and‎ ‎give an interesting identity for it.

An Estimate of Incomplete Mixed Character Sums

Upper bound estimate of character sums over Lehmer's numbers

*Correspondence: [email protected] Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R. China Abstract Let p be an odd prime. For each integer awith 1≤ a≤ p – 1, it is clear that there exists one and only one awith 0≤ a≤ p – 1 such that a · a≡ 1 mod p. LetA denote the set of all integers 1≤ a≤ p – 1, in which a and a are of opposite parity. The main purpose of this paper is us...

Mean Value Estimates for Odd Cubic Weyl Sums

We establish an essentially optimal estimate for the ninth moment of the exponential sum having argument αx + βx. The first substantial advance in this topic for over 60 years, this leads to improvements in Heath-Brown’s variant of Weyl’s inequality, and other applications of Diophantine type.

Near-optimal Mean Value Estimates for Multidimensional Weyl Sums

We obtain sharp estimates for multidimensional generalisations of Vinogradov’s mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed.