共查询到20条相似文献,搜索用时 15 毫秒
1.
Aleksandar Ivić 《The Ramanujan Journal》2009,19(2):207-224
We obtain, for T
ε
≤U=U(T)≤T
1/2−ε
, asymptotic formulas for
where Δ(x) is the error term in the classical divisor problem, and E(T) is the error term in the mean square formula for
. Upper bounds of the form O
ε
(T
1+ε
U
2) for the above integrals with biquadrates instead of square are shown to hold for T
3/8≤U=U(T)≪
T
1/2. The connection between the moments of E(t+U)−E(t) and
is also given. Generalizations to some other number-theoretic error terms are discussed.
相似文献
2.
Matti Jutila 《Arkiv f?r Matematik》1983,21(1-2):75-96
3.
Kai-Man Tsang 《中国科学 数学(英文版)》2010,53(9):2561-2572
Let Δ(x) and E(t) denote respectively the remainder terms in the Dirichlet divisor problem and the mean square formula for the Riemann zeta-function on the critical line.This article is a survey of recent developments on the research of these famous error terms in number theory.These include upper bounds,Ω-results,sign changes,moments and distribution,etc.A few open problems are also discussed. 相似文献
4.
周焕芹 《纯粹数学与应用数学》2008,24(1):41-44
对任意正整数n,著名的Smarandache函数S(n)定义为最小的正整数m使得n|m!.即S(n)=min{m∶m ∈N,n|m!).本文的主要目的是利用初等方法研究一类包含S(n)的Dirichlet级数与Riemann zeta-函数之间的关系,并得到了一个有趣的恒等式. 相似文献
5.
Yuk-Kam Lau 《Monatshefte für Mathematik》1994,117(1-2):103-106
Ifu
n
denotes thenth zero of the function
,Ivi has shown thatu
n+1
–u
n
u
n
1/2
for alln andu
n+1
–u
n
u
n
1/2
(log un)–5for infinitely manyn. We sharpen his lower estimate for the gapu
n+1
–u
n
o the best possible, namely,u
n+1
–u
n
u
n
1/2
for infinitely manyn.The author wishes to thank Dr. Kai-Man Tsang for his continual guidance. 相似文献
6.
7.
Aleksandar Ivić 《Central European Journal of Mathematics》2004,2(4):494-508
Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of
. If
with
, then we obtain
. We also show how our method of proof yields the bound
, where T
1/5+ε≤G≪T, T<t
1<...<t
R
≤2T, t
r
+1−t
r
≥5G (r=1, ..., R−1). 相似文献
8.
9.
For a positive integer n we let τ(n) denote the number of its positive divisors. In this paper, we obtain lower and upper bounds for the average value of the
ratio τ(n + 1)/τ(n) as n ranges through positive integers in the interval [1,x]. We also study the cardinality of the sets {τ(p − 1) : p ≤ x prime} and {τ(2n − 1) : n ≤ x}.
Authors’ addresses: Florian Luca, Instituto de Matemáticas, Universidad Nacional Autónoma'de'México, C.P. 58089, Morelia,
Michoacán, México; Igor E. Shparlinski, Department of Computing, Macquarie University, Sydney, NSW 2109, Australia 相似文献
10.
11.
关于k次加法补函数的因子函数的均值公式 总被引:1,自引:0,他引:1
对于任意正整数n,如果m n是完全k次方数,称最小非负整数m是n的k次加法补.为了研究m的性质及变化规律,这里运用初等数论和分析数论的方法,得到了d(n ak(n))的一个有趣的均值公式,从而得到了更一般的加法补函数的计算公式,完善了加法补函数在数论中的研究和应用. 相似文献
12.
Aleksandar Ivić 《Central European Journal of Mathematics》2005,3(2):203-214
Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of
. If E
*(t)=E(t)-2πΔ*(t/2π) with
, then we obtain
and
It is also shown how bounds for moments of | E
*(t)| lead to bounds for moments of
. 相似文献
13.
A. Laurinčikas 《Lithuanian Mathematical Journal》1995,35(4):399-402
The research has been partially supported by Grant N LAC000 from the International Science Foundation. 相似文献
14.
15.
16.
Markus Niess 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2009,44(5):335-339
The Riemann zeta-function ζ has the following well-known properties
(M) It is meromorphic in ℂ with a simple pole at z = 1 with residue 1. 相似文献
17.
18.
In this article we study two problems raised by a work of Conrey and Ghosh from 1989. Let ζ(k)(s) be the k-th derivative of the Riemann zeta-function, and χ(s) be factor in the functional equation of the Riemann zeta-function. We calculate the average values of ζ(j) and χ at the nontrivial zeros of ζ(k). 相似文献
19.
For 1/2<<1 fixed, letE
(T) denote the error term in the asymptotic formula for
. We obtain some new bounds forE
(T), and an _-result which is the analogue of the strongest _-result in the classical Dirichlet divisor problem. 相似文献
20.