On character sums over a short interval

Weighted Short-interval Character Sums

In this paper we shall establish the counterpart of Szmidt, Urbanowicz and Zagier’s formula in the sense of the Hecker correspondence. The motivation is the derivation of the values of the Riemann zeta-function at positive even integral arguments from the partial fraction expansion for the hyperbolic cotangent function (or the cotangent function). Since the last is equivalent to the functional ...

On Certain Character Sums over Smooth Numbers

Character sums over norm groups

In this paper, we give estimates of character sums of Weil type. They are associated with polynomials and rational functions with coefficients in a quadratic extension k2 of a finite field k of q elements, additive and multiplicative characters of k2, and summed over the kernel of the norm map from k× 2 to k×. The bounds are of the magnitude O( √ q) with the implied constant depending on the de...

Short Character Sums with Fermat Quotients

The distribution of Fermat quotients and related sequences is interesting from several perspectives. First, there are several applications, in particular to algebraic number theory and computer science. Fermat quotients play for instance a role in primality testing (see [L]) and are well-studied as model for generating pseudo-random numbers. (See [COW].) From the analytical side, establishing d...

Short Character Sums with Beatty Sequences

Abstract. We estimate multiplicative character sums taken on the values of a nonhomogeneous Beatty sequence {⌊αn+ β⌋ : n = 1, 2, . . . }, where α, β ∈ R, and α is irrational. In particular, our bounds imply that for fixed α, β and a small real number ε > 0, if p is sufficiently large and p1/3+ε ≤ N ≤ p1/2+ε, then among the first N elements of the Beatty sequence there are N/2+ o(N) quadratic no...