An estimate for character sums

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...

Estimates for Mixed Character Sums

a nontrivial additive character of k. We are given a polynomial f = f(x1, ..., xn) in n ≥ 1 variables over k of degree d ≥ 1 which is a “Deligne polynomial”, i.e., its degree d is prime to p and its highest degree term, say fd, is a homogeneous form of degree d in n variables which is nonzero, and whose vanishing, if n ≥ 2, defines a smooth hypersurface in the projective space Pn−1. For a Delig...

Large Character Sums

where χ is a non-principal Dirichlet character χ (mod q). It is easy to show that such character sums are always ≤ q in absolute value, while G. Pólya and I.M. Vinogradov (see [3]) improved this to≤ √q log q around 1919, and H.L. Montgomery and R.C. Vaughan [13] to √q log log q in 1977, assuming the Generalized Riemann Hypothesis (GRH). Up to the constant this is “best possible” since R.E.A.C. ...

Hermite Character Sums

This evaluation proves a conjecture in [9, p. 370] and solves the problem of finding explicitly the number of rational points (mod/?) on the surface z = (x + l)(y + ΐ)(x + y), a problem some algebraic geometers had worked on without success. Character sum analogues of the important formulas for orthogonal polynomials are potentially as useful as those for hypergeometric series, so a systematic ...