Zhang Wenpeng

On the general Gauss sums and their fourth power mean

Lfeing the elementary method the calculating problem of the fourth power mean of the general k-th Gauss sums is studied,and an exact calculating formula is given.

A MEAN VALUE RELATED TO D. H. LEHMER'S PROBLEM AND THE RAMANUJAN'S SUM

Let q > 1 be an odd integer and c be a fixed integer with ( c, q ) = 1. For each integer a with 1 ≤ a ≤ q − 1, it is clear that there exists one and only one b with 0 ≤ b ≤ q − 1 such that ab ≡ c (mod q ). Let N ( c, q ) denotes the number of all solutions of the congruence equation ab ≡ c (mod q ) for 1 ≤ a, b ≤ q − 1 in which a and b are of opposite parity, where b is defined by the congruence equation b b ≡ 1(mod q ). The main purpose of this paper is using the mean value theorem of Dirichlet L -functions to study the mean value properties of a summation involving ( N ( c, q ) − φ( q )) and Ramanujans sum, and give two exact computational formulae.

The ternary exponential Diophantine equation concerning Pythagorean triplets

Abstract Let r be a positive integer with r > 1, and let m be a positive even integer. Further let a=|V(m, r)|, b=|U (m, r)| and c=m2+1, where V (m, r) + U (m, r)√−1=(m+√−1)r . In this paper, using the Gel′ fond-Baker method, we prove that if 2 | r and m>max(1015, 2r3), then the equation ax+by=cz has only the positive integer solution (x, y, z)=(2, 2, r).

Some Applications of L-Functions to the Mean Value of the Dedekind Sums and Cochrane Sums

In this paper, the applications of L- functions to the mean value of Dedekind sums and Cochrane sums are described, and a few asymptotic formulae are presented.

On the mean value of the square of a generalized Dedekind sum

The main purpose of this paper is to use the mean value theorem of the Dirichlet L-functions to study the distribution property of a generalized Dedekind sum, and give a sharper mean square value formula.