Tianping Zhang

A generalization on the difference between an integer and its inverse modulo

Let q > 2 and c are two integers with (q, c) = 1, for each integer a with 0 2 we have M(q,k,c,n)=Φ n (q)q 2 k n (2k+1) n + O (4 k q ( 2 k + 1 ) n - ( 1 / 2 ) d 2 (q)lnq).

q

Abstract Let q ≥ 2 be an integer, χ be a non-principal character mod q, A = A(q) ≤ q, B = B(q) ≤ q, and H = H(q) ≤ q. In this paper, we shall use the estimates for Kloosterman sums and the properties of trigonometric sums to give some sharp estimates for character sums of the form where ħ(A, B, H) = {a ∈ ℤ| (a, q) = 1, 1 ≤ a ≤ A, 1 ≤ b ≤ B, ab ≡ 1 (mod q), |a − b| ≤ H}.

. (II)

Abstract By using the properties of Kloosterman sum and Dirichlet character, an optimal upper bound estimate of incomplete Cochrane sum is given.

On certain character sums

The high-dimensional D. H. Lehmer problem over quarter intervals is studied. By using the properties of character sum and the estimates of Dirichlet -function, the previous result is improved to be the best possible in the case of , an odd prime with (mod 4), which is shown by studying the mean square value of the error term.