On a problem of D.H. Lehmer over short intervals

Let p be an odd prime and a be an integer coprime to p. Denote by N(a,p) the number of pairs of integers b,c with bc≡a (mod p), and with b,c having different parity. The main purpose of this paper is to study the sum , and obtain a sharp asymptotic formula.     

On a Problem of D. H. Lehmer and General Kloosterman Sums

Abstract Let q ≥ 3 be an odd number, a be any fixed positive integer with (a, q) = 1. For each integer b with 1 ≤ b < q and (b, q) = 1, it is clear that there exists one and only one c with 0 < c < q such that bc ≡ a (mod q). Let N(a, q) denote the number of all solutions of the congruent equation bc ≡ a (mod q) for 1 ≤ b, c < q in which b and c are of opposite parity, and let . The main purpose of this paper is to study the distribution properties of E(a, q), and give a sharper hybrid mean-value formula involving E(a, q) and general Kloosterman sums. This work is supported by the NSF and the PSF of P. R. China     

On the generalization of the D. H. Lehmer problem

Let n ≥ 2 be a fixed positive integer, q ≥ 3 and c be two integers with （n, q） = （c, q） = 1. We denote by rn（51, 52, C; q） （δ 〈 δ1,δ2≤ 1） the number of all pairs of integers a, b satisfying ab ≡ c（mod q）, 1 〈 a ≤δ1q, 1 ≤ b≤δ2q, （a,q） = （b,q） = 1 and nt（a＋b）. The main purpose of this paper is to study the asymptotic properties of rn （δ1, δ2, c; q）, and give a sharp asymptotic formula for it.     

A problem of D.H. Lehmer and its mean square value formula

Let q be an odd positive integer and let a be an integer coprime to q. For each integer b coprime to q with 1?b<q, there is a unique integer c coprime to q with 1?c<q such that . Let N(a,q) denote the number of solutions of the congruence equation with 1?b,c<q such that b,c are of opposite parity. The main purpose of this paper is to use the properties of Dedekind sums, the properties of Cochrane sums and the mean value theorem of Dirichlet L-functions to study the asymptotic property of the mean square value , and give a sharp asymptotic formula.     

Kloosterman Sums and a Problem of D. H. Lehmer

A classical problem of D. H. Lehmer suggests the study of distributions of elements of Z/pZ of opposite parity to the multiplicative inverse mod p. Zhang initiated this problem and found an asymptotic evaluation of the number of such elements. In this paper, an asymptotic formula for the fourth moment of the error term of Zhang is proved,from which one may see that Zhang’s error term is optimal up to the logarithm factor.The method also applies to the case of arbitrary positive integral moments.     

High-dimensional D. H. Lehmer problem over short intervals

Letk be a positive integer and n a nonnegative integer,0 λ1,...,λk+1 ≤ 1 be real numbers and w =(λ1,λ2,...,λk+1).Let q ≥ max{[1/λi ]:1 ≤ i ≤ k + 1} be a positive integer,and a an integer coprime to q.Denote by N(a,k,w,q,n) the 2n-th moment of(b1··· bk c) with b1··· bk c ≡ a(mod q),1 ≤ bi≤λiq(i = 1,...,k),1 ≤ c ≤λk+1 q and 2(b1+ ··· + bk + c).We first use the properties of trigonometric sum and the estimates of n-dimensional Kloosterman sum to give an interesting asymptotic formula for N(a,k,w,q,n),which generalized the result of Zhang.Then we use the properties of character sum and the estimates of Dirichlet L-function to sharpen the result of N(a,k,w,q,n) in the case ofw =(1/2,1/2,...,1/2) and n = 0.In order to show our result is close to the best possible,the mean-square value of N(a,k,q) φk(q)/2k+2and the mean value weighted by the high-dimensional Cochrane sum are studied too.     

关于D.H.Lehmer问题与超级Kloosterman和

设整数q>2,c与q互素．对于1到q之间与q互素的任意整数a,在1到q之间存在唯一的整数b满足ab≡c mod q．对任意整数k≥2,定义M(q,k,c)为满足1≤ai≤q, (ai,q)=1,i=1,2,…,k,a1a2…ak≡c mod q且2 a1 a2 … ak的正整数组(a1,a2,…,ak)的数目,并设E(q,k,c)=M(q,k,c)-(φk-1(q))/2．本文的主要目的是利用Gauss和与原特征的性质,以及Dirichlet L-函数的均值定理,来研究E(q,k,c)与超级Kloosterman和K(h,k,q)的混合均值,并给出一个均值公式．     

On a Problem of D. H. Lehmer

The main purpose of this paper is to use the Fourier expansion for character sums and the mean value theorem of Dirichlet L-functions to study the asymptotic property of the difference between a D. H. Lehmer number and its inverse modulo p （an odd prime）, A interesting mean square value formula is also given.     

A note on the Lehmer problem over short intervals

Let p be an odd prime, c be an integer with (c, p) = 1, and let N be a positive integer with N ≤ p − 1. Denote by r(N, c; p) the number of integers a satisfying 1 ≤ a ≤ N and 2 ∤ a + ā, where ā is an integer with 1 ≤ ā ≤ p − 1, aā ≡ c (mod p). It is well known that r(N, c; p) = 1/2N + O(p ^1/2log² p). The main purpose of this paper is to give an asymptotic formula for Σ _c=1 ^p−1(r(N, c; p) − 1/2N)².     

广义Dedekind函数误差项的加权平方积分均值

研究了和式∑↑n≤x（n/ψ（n））^r的渐近公式中的误差项E（x;r）的加权平方积分均值。其中，r是正整数，ψ(n)是Dedekind函数。     

一个新的算术函数及其均值

对任意正整数n,我们定义算术函数(Ω)(n)为(Ω)(1)=0,当n＞1,且n=pα11·pα22…pαkk为n的标准分解式时,定义(Ω)(n)=α1p1 α2p2 … αkpk.显然这个函数是可加函数.即就是对任意正整数m及n有(Ω)(m·n)=(Ω)(m) (Ω)(n).本文主要目的是利用初等方法研究函数(Ω)(n)的算术性质,并给出一个较强的均值公式及有趣的恒等式.     

On a problem of D. H. Lehmer

Let be an odd prime number. For we denote the inverse of modulo by with . Given , we prove that in any range of length the probability that has the same parity as tends to as . This result was previously known only to hold true in the full range of length . We will also obtain quantitative results on the pseudorandomness of the sequence for which we estimate the well-distribution and correlation measures as defined by Mauduit and Sárközy (1997).

     

一个新的Smarandache函数及其均值

对任意正整数n≥3,我们定义算术函数C(n)为最大的正整数m≤n-2使得n |Cnm=n!/m!·(n-m)!.即就是C(n)=max{m:m≤n-2,n|Cnm},并规定C(1)=C(2)=1.本文的主要目的是利用初等及解析方法研究这一函数的均值分布问题,并给出几个有趣的均值公式及渐近式.     

一个新的数论函数及其均值性质

对于给定的自然数q,我们利用最小公倍数定义一个新的数论函数g(n)=[q,n]/q.本文的主要目的是研究δm(g(n))的均值性质,并利用解析方法得到这个函数的几个渐近公式.     

一个数论函数的p次均值的计算

主要解决了二进制数字之和函数的p次均值的计算问题,对二进制数字之和函数的p次均值的计算进行了猜想,归纳,得出了精确的p次计算公式Ap(N).     

一个新的数论函数及其均值

设f(n)为任一数论函数,本文的主要目的是引入一类新的可乘函数g f(n),其定义如下:当n=1时,g f(1)=1;对任意素数p及正整数α≥1,定义g f(pα)=pf(α),并给出其均值的两个有趣的渐近公式.     

整数分解式的指数函数及其均值的计算

给出了整数标准分解式的数论函数β(n，p)和它的均值∑n≤Nβ(n，p)的定义，推出了整数n标准分解式的指数函数计算公式和均值∑n≤Nβ(n,p)的精确计算公式．     

On a problem of D.H. Lehmer and pseudorandom binary sequences

Let p be an odd prime, and f(x), g(x) ∈ [x]. Define

A mean value of Cochrane sum

Let q 〉 4 be an integer. The main purpose of this paper is to study the mean value of Cochrane sum C（a, q） in quarter intervals, and obtain a sharp asymptotic formula for it.     

A mean value theorem on Dirichlet series

This paper deals with mean-value for the square of certain functionF(s) which has some characteristic properties of the Riemann zeta-function and its powers.Work supported by the University of the Basque Country.