共查询到20条相似文献,搜索用时 15 毫秒
1.
Danilo Bazzanella 《Archiv der Mathematik》2011,97(5):453-458
Let d(n) denote the number of positive divisors of the natural number n. The aim of this paper is to investigate the validity of the asymptotic formulafor \({x \to + \infty,}\) assuming a hypothetical estimate on the meanwhich is a weakened form of a conjecture of M. Jutila.
相似文献
$\begin{array}{lll}\sum \limits_{x < n \leq x+h(x)}d(n)\sim h(x)\log x\end{array}$
$\begin{array}{lll} \int \limits_X^{X+Y}(\Delta(x+h(x))-\Delta (x))^2\,{d}x, \end{array}$
2.
3.
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
4.
E. I. Panteleeva 《Mathematical Notes》1988,44(4):750-757
5.
In this paper we establish an asymptotic formula for the sum
when y is large compared to x1/2 log x.
Received: 27 January 2005 相似文献
6.
In this paper we consider the Dirichlet series obtained from the error term in the Dirichlet divisor problem. We shall prove their analytic continuations and determine the locations of poles. We shall also discuss the relation between our results and the conjecture of Lau and Tsang on the mean square estimate of the error term. 相似文献
7.
Jun Furuya 《Journal of Number Theory》2005,115(1):1-26
Let Δ(x) be the error term in the Dirichlet divisor problem. The purpose of this paper is to study the difference between two kinds of mean value formulas of Δ(x), that is, the mean value formulas and ∑n?xΔ(n)k with a natural number k. In particular we study the case k=2 and 3 in detail. 相似文献
8.
In this paper we prove (in a rather more precise form) two conjectures of P. Erdös about the maximum and minimum values of the divisor function on intervals of length k. 相似文献
9.
Archiv der Mathematik - 相似文献
10.
Ubertino Battisti 《Mathematische Zeitschrift》2012,272(3-4):1365-1381
We consider a class of pseudodifferential operators, with crossed vector valued symbols, defined on the product of two closed manifolds. We study the asymptotic expansion of the counting function of positive selfadjoint operators in this class. Using a general Theorem of Aramaki, we can determine the first term of the asymptotic expansion of the counting function and, in a special case, we are able to find the second term. We give also some examples, emphasizing connections with problems of analytic number theory, in particular with Dirichlet divisor function. 相似文献
11.
A. I. Pavlov 《Mathematical Notes》2000,68(3):370-377
The main result of this paper is the following theorem. Suppose thatτ(n) = ∑
d|n
l and the arithmetical functionF satisfies the following conditions:
Then there exist constantsA
1,A
2, andA
3 such that for any fixed \g3\s>0 the following relation holds:
. Moreover, if for any primep the inequality \vbf(p)\vb\s<1 holds and the functionF is strongly multiplicative, thenA
1\s>0.
Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 429–438, September, 2000. 相似文献
1) | the functionF is multiplicative; |
2) | ifF(n) = ∑ d|n f(d), then there exists an α>0 such that the relationf(n)=O(n −α) holds asn→∞. |
12.
13.
To ProfessorEkkehard Kr?tzel on his 60th birthday 相似文献
14.
We obtain a new upper bound for the sum Σ
h≤H
Δ
k
(N, h) when 1 ≤ H ≤ N, k ∈ ℕ, k ≥ 3, where Δ
k
(N, h) is the (expected) error term in the asymptotic formula for Σ
N<n≤2N
d
k
(n)d
k
(n + h), and d
k
(n) is the divisor function generated by ζ(s)
k
. When k = 3, the result improves, for H ≥ N
1/2, the bound given in a recent work of Baier, Browning, Marasingha and Zhao, who dealt with the case k = 3. 相似文献
15.
Kaisa Matomäki 《Monatshefte für Mathematik》2008,155(2):167-189
Let 1/5 < θ ≤ 1. We prove that there exists a positive constant δ such that the number of even integers in the interval [X, X + X
θ] which are not a sum of two primes is 《 X
θ−δ. The proof uses the circle method, a sieve method, exponential sum estimates and zero-density estimates for L-functions.
Current address: Department of Mathematics, 20014 University of Turku, Finland.
Author’s address: Department of Mathematics, University of London, Royal Holloway, Egham, Surrey TW20 0EX, UK 相似文献
16.
José Villa-Morales 《Expositiones Mathematicae》2012,30(4):406-411
A particular case of the Dirichlet problem is solved using the Convergence Theorem for discrete-time martingales and the mean value property of harmonic functions as the main tools. 相似文献
17.
O. V. Kolpakova 《Mathematical Notes》2011,89(3-4):504-518
We obtain a new value of the Karatsuba constant in the multidimensional Dirichlet divisor problem. We also find a new value of the exponent of the main parameter in the estimate of the mean value of the remainder in a given asymptotics. The proof of the main statements is based on the derivation of a new estimate of the Carleson abscissa in the theory of the Riemann zeta function. 相似文献
18.
O. V. Kolpakova 《Moscow University Mathematics Bulletin》2007,62(6):244-246
An estimate of the remainder term in an asymptotic formula for Riesz means of the multidimensional divisor function is obtained with an arbitrary value of the parameter α > 0. 相似文献
19.
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
20.
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