Asymptotic Formulae for Pairs of Diagonal Cubic Equations

2011 ◽  
Vol 63 (1) ◽  
pp. 38-54 ◽  
Author(s):  
Jörg Brüdern ◽  
Trevor D. Wooley
Keyword(s):  
Asymptotic Formula ◽  
Linear Combination ◽  
Asymptotic Formulae ◽  
Local Densities ◽  
Integral Solutions

Abstract We investigate the number of integral solutions possessed by a pair of diagonal cubic equations in a large box. Provided that the number of variables in the system is at least fourteen, and in addition the number of variables in any non-trivial linear combination of the underlying forms is at least eight, we obtain an asymptotic formula for the number of integral solutions consistent with the product of local densities associated with the system.

scholarly journals Explicit and asymptotic formulae for Vasyunin-cotangent sums

2017 ◽  
Vol 102 (116) ◽  
pp. 155-174 ◽  
Author(s):  
Mouloud Goubi ◽  
Abdelmejid Bayad ◽  
Mohand Hernane
Keyword(s):  
Asymptotic Formula ◽  
Explicit Formula ◽  
Continued Fraction ◽  
Reciprocity Law ◽  
Asymptotic Formulae

For coprime numbers p and q, we consider the Vasyunin-cotangent sum V(q, p)= ?p?1 k=1 {kq/p} cot (?k/p). First, we prove explicit formula for the symmetric sum V(p,q)+ V(q,p) which is a new reciprocity law for the sums above. This formula can be seen as a complement to the Bettin-Conrey result [13, Theorem 1]. Second, we establish an asymptotic formula for V(p,q). Finally, by use of continued fraction theory, we give a formula for V(p,q) in terms of continued fraction of p/q.

scholarly journals SOME ESTIMATES OF SPECIAL CLASSES OF INTEGRALS

2000 ◽  
Vol 5 (1) ◽  
pp. 127-132
Author(s):  
T. I. Malyutina
Keyword(s):  
Asymptotic Formula ◽  
Asymptotic Expansions ◽  
Entire Functions ◽  
Regular Growth ◽  
Asymptotic Formulas ◽  
Arbitrary Parameter ◽  
Subharmonic Functions ◽  
Completely Regular ◽  
Asymptotic Formulae ◽  
New Variant

We study the integrals fb a f(t) exp(i| ln rt|σ) dt and obtain asymptotic formula for these functions of non‐regular growth. This is a peculiar kind of the theory asymptotic expansions. In particular, we get asymptotic formulae for different entire functions of non‐regular growth. Asymptotic formulas for Levin‐Pfluger entire functions of completely regular growth are well‐known [1]. Our formulas allow to find limiting Azarin's [2] sets for some subharmonic functions. The kernel exp(i| ln rt|σ) contains arbitrary parameter σ > 0. The integrals for σ ∈(0, 1), σ = 1, σ > 1 essentially differ. Our arguments can apply to more general kernels. We give a new variant of the classic lemma of Riemann and Lebesgue from the theory of the transformation of Fourier.

On Waring's problem for cubes and biquadrates. II

1988 ◽  
Vol 104 (2) ◽  
pp. 199-206 ◽  
Author(s):  
Jörg Brüdern
Keyword(s):  
Lower Bound ◽  
Asymptotic Formula ◽  
Riemann Hypothesis ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Asymptotic Formulae ◽  
Sixth Moment ◽  
Almost All

In discussing the consequences of a conditional estimate for the sixth moment of cubic Weyl sums, Hooley [4] established asymptotic formulae for the number ν(n) of representations of n as the sum of a square and five cubes, and for ν(n), defined similarly with six cubes and two biquadrates. The condition here is the truth of the Riemann Hypothesis for a certain Hasse–Weil L-function. Recently Vaughan [8] has shown unconditionally , a lower bound of the size suggested by the conditional asymptotic formula. In the corresponding problem for ν(n) the author [1] was able to deduce ν(n) > 0, as a by-product of the result that almost all numbers can be expressed as the sum of three cubes and one biquadrate. As promised in the first paper of this series we return to the problem of bounding ν(n) from below.

scholarly journals On two sums related to the Lehmer problem over short intervals

2021 ◽  
Vol 6 (11) ◽  
pp. 11723-11732
Author(s):  
Yanbo Song ◽  
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Short Intervals ◽  
Asymptotic Formulae ◽  
Lehmer Problem ◽  
Lehmer Numbers

<abstract><p>In this article, we study sums related to the Lehmer problem over short intervals, and give two asymptotic formulae for them. The original Lehmer problem is to count the numbers coprime to a prime such that the number and the its number theoretical inverse are in different parities in some intervals. The numbers which satisfy these conditions are called Lehmer numbers. It prompts a series of investigations, such as the investigation of the error term in the asymptotic formula. Many scholars investigate the generalized Lehmer problems and get a lot of results. We follow the trend of these investigations and generalize the Lehmer problem.</p></abstract>

Asymptotic formulae in the theory of partitions

1954 ◽  
Vol 50 (2) ◽  
pp. 225-241 ◽  
Author(s):  
C. B. Haselgrove ◽  
H. N. V. Temperley
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Classical Method ◽  
Contour Integration ◽  
Large Positive Integer ◽  
Asymptotic Formulae ◽  
Positive Integers

It is the object of this paper to obtain an asymptotic formula for the number of partitions pm(n) of a large positive integer n into m parts λr, where the number m becomes large with n and the numbers λ1, λ2,… form a sequence of positive integers. The formula is proved by using the classical method of contour integration due to Hardy, Ramanujan and Littlewood. It will be necessary to assume certain conditions on the sequence λr, but these conditions are satisfied in most of the cases of interest. In particular, we shall be able to prove the asymptotic formula in the cases of partitions into positive integers, primes and kth powers for any positive integer k.

scholarly journals On the average order of the gcd-sum function over arbitrary sets of integers

2021 ◽  
Vol 27 (3) ◽  
pp. 16-28
Author(s):  
V. Siva Rama Prasad ◽  
◽  
P. Anantha Reddy ◽  
Keyword(s):  
Asymptotic Formula ◽  
Greatest Common Divisor ◽  
Average Order ◽  
Summatory Function ◽  
Asymptotic Formulae ◽  
Error Terms ◽  
Positive Integers

Let \mathbb{N} denote the set of all positive integers and for j,n \in \mathbb{N}, let (j,n) denote their greatest common divisor. For any S\subseteq \mathbb{N}, we define P_{S}(n) to be the sum of those (j,n) \in S, where j \in \{1,2,3, \ldots, n\}. An asymptotic formula for the summatory function of P_{S}(n) is obtained in this paper which is applicable to a variety of sets S. Also the formula given by Bordellès for the summatory function of P_{\mathbb{N}}(n) can be derived from our result. Further, depending on the structure of S, the asymptotic formulae obtained from our theorem give better error terms than those deducible from a theorem of Bordellès (see Remark 4.4).

scholarly journals SHORT INTERVALS ASYMPTOTIC FORMULAE FOR BINARY PROBLEMS WITH PRIME POWERS, II

2019 ◽  
Vol 109 (3) ◽  
pp. 351-370 ◽  
Author(s):  
ALESSANDRO LANGUASCO ◽  
ALESSANDRO ZACCAGNINI
Keyword(s):  
Asymptotic Formula ◽  
Prime Numbers ◽  
Short Intervals ◽  
Asymptotic Formulae ◽  
Representations Of Integers ◽  
Binary Problems

AbstractWe improve some results in our paper [A. Languasco and A. Zaccagnini, ‘Short intervals asymptotic formulae for binary problems with prime powers’, J. Théor. Nombres Bordeaux30 (2018) 609–635] about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell _{1}}+p_{2}^{\ell _{2}}$ and $n=p^{\ell _{1}}+m^{\ell _{2}}$, where $\ell _{1},\ell _{2}\geq 2$ are fixed integers, $p,p_{1},p_{2}$ are prime numbers and $m$ is an integer. We also remark that the techniques here used let us prove that a suitable asymptotic formula for the average number of representations of integers $n=\sum _{i=1}^{s}p_{i}^{\ell }$, where $s$, $\ell$ are two integers such that $2\leq s\leq \ell -1$, $\ell \geq 3$ and $p_{i}$, $i=1,\ldots ,s$, are prime numbers, holds in short intervals.

scholarly journals Notes on a General Sequence

2020 ◽  
Vol 34 (2) ◽  
pp. 193-202
Author(s):  
Reza Farhadian ◽  
Rafael Jakimczuk
Keyword(s):  
Asymptotic Formula ◽  
Counting Function ◽  
Real Numbers ◽  
General Sequence ◽  
Asymptotic Formulae

AbstractLet {rn}n∈𝕅 be a strictly increasing sequence of nonnegative real numbers satisfying the asymptotic formula rn ~ αβn, where α, β are real numbers with α > 0 and β > 1. In this note we prove some limits that connect this sequence to the number e. We also establish some asymptotic formulae and limits for the counting function of this sequence. All of the results are applied to some well-known sequences in mathematics.

A scaling approach to bumps and multi-bumps for nonlinear partial differential equations

2006 ◽  
Vol 136 (3) ◽  
pp. 585-614 ◽  
Author(s):  
Robert Magnus
Keyword(s):  
Hilbert Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Asymptotic Formula ◽  
Linear Combination ◽  
Implicit Function Theorem ◽  
Nonlinear Partial Differential Equations ◽  
Implicit Function ◽  
Space Operator ◽  
Norm Topology

The problem −Δu + F(V (εx), u) = 0 is considered in Rn. For small ε > 0, solutions are obtained that approach, as ε → 0, a linear combination of specified functions, mutually translated by O(1/ε). These are the so-called multi-bump solutions. The method involves a rescaling of the variables and the use of a modified implicit function theorem. The usual implicit function theorem is inapplicable, owing to lack of convergence of the derivative of the nonlinear Hilbert space operator, obtained after an appropriate rescaling, in the operator-norm topology. An asymptotic formula for the solution for small ε is obtained.

Heavy Tails of Discounted Aggregate Claims in the Continuous-Time Renewal Model

2007 ◽  
Vol 44 (02) ◽  
pp. 285-294 ◽  
Author(s):  
Qihe Tang
Keyword(s):  
Asymptotic Formula ◽  
Continuous Time ◽  
Heavy Tails ◽  
Tail Behavior ◽  
Renewal Model ◽  
Discounted Aggregate Claims ◽  
Aggregate Claims ◽  
Tail Asymptotic

We study the tail behavior of discounted aggregate claims in a continuous-time renewal model. For the case of Pareto-type claims, we establish a tail asymptotic formula, which holds uniformly in time.

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