研究了一个与Weil估计有关的等式.由于广义的二次和有很好的均值性质,近几年,有许多文章都在研究广义二次高斯和的均值.用初等数论的方法把文献(J.Num.Theory,2009,129(4):1075-1089.)中的第一个定理中的数1推广到任意的满足(w,p)=1的w,并且得到了如下一般的结果sum from u=1 to (p-1)sum from v=1 to (p-1)((u^2-v^2)/p)((v^2-w)/p)=(1+((-1)/p))(1+((-w)/p)),p■w.
Let ,4 = {1 ≤ a1 〈 a2 〈 ...} be a sequence of integers. ,4 is called a sum-free sequence if no ai is the sum of two or more distinct earlier terms. Let A be the supremum of reciprocal sums of sum-free sequences. In 1962, ErdSs proved that A 〈 103. A sum-free sequence must satisfy an ≥ （k ~ 1）（n - ak） for all k, n ≥ 1. A sequence satisfying this inequality is called a x-sequence. In 1977, Levine and O＇Sullivan proved that a x-sequence A with a large reciprocal sum must have al = 1, a2 = 2, and a3 = 4. This can be used to prove that λ 〈 4. In this paper, it is proved that a x-sequence A with a large reciprocal sum must have its initial 16 terms： 1, 2, 4, 6, 9, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, and 50. This together with some new techniques can be used to prove that λ 〈 3.0752. Three conjectures are posed.
For a set A of nonnegative integers, the representation functions R2(A,n) and R3(A,n) are defined as the numbers of solutions to the equation n = a + a′ with a,a′∈ A, a < a′ and a a′, respectively. Let N be the set of nonnegative integers. Given n0 > 0, it is known that there exist A,A′■ N such that R2(A′,n) = R2(N \ A′,n) and R3(A,n) = R3(N \ A,n) for all n n0. We obtain several related results. For example, we prove that: If A ■ N such that R3(A,n) = R3(N \ A,n) for all n n0, then (1) for any n n0 we have R3(A,n) = R3(N \ A,n) > c1n - c2, where c1,c2 are two positive constants depending only on n0; (2) for any α < 116, the set of integers n with R3(A,n) > αn has the density one. The answers to the four problems in Chen-Tang (2009) are affirmative. We also pose two open problems for further research.
For any given coprime integers p and q greater than 1, in 1959, B proved that all sufficiently large integers can be expressed as a sum of pairwise terms of the form p^aq^b. As Davenport observed, Birch＇s proof can be modified that the exponent b can be bounded in terms of p and q. In 2000, N. Hegyvari effective version of this bound. The author improves this bound.
