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).

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 Sums of weighted averages of gcd-sum functions

2018 ◽  
Vol 14 (10) ◽  
pp. 2699-2728 ◽  
Author(s):  
Isao Kiuchi ◽  
Sumaia Saad eddin
Keyword(s):  
Mean Value ◽  
Arithmetical Function ◽  
Partial Sums ◽  
Asymptotic Formulas ◽  
Greatest Common Divisor ◽  
Fixed Integer ◽  
Weighted Averages ◽  
Error Terms ◽  
Positive Integers

Let [Formula: see text] be the greatest common divisor of the integers [Formula: see text] and [Formula: see text]. In this paper, we give several interesting asymptotic formulas for weighted averages of the [Formula: see text]-sum function [Formula: see text] and the function [Formula: see text] for any positive integers [Formula: see text] and [Formula: see text], namely [Formula: see text] with any fixed integer [Formula: see text] and any arithmetical function [Formula: see text]. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of [Formula: see text]-sum functions [Formula: see text]

XLIX.—Generalizations of a Problem of Pillai

1949 ◽  
Vol 62 (4) ◽  
pp. 460-469
Author(s):  
L. Mirsky
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Main Concern ◽  
Positive Integers ◽  
Infinite Set

I. Throughout this paper k1, …, k3 will denote s ≥ I fixed distinct positive integers. Some years ago Pillai (1936) found an asymptotic formula, with error term O(x/log x), for the number of positive integers n ≤ x such that n + k1, …, n + k3 are all square-free. I recently considered (Mirsky, 1947) the corresponding problem for r-free integers (i.e. integers not divisible by the rth power of any prime), and was able, in particular, to reduce the error term in Pillai's formula.Our present object is to discuss various generalizations and extensions of Pillai's problem. In all investigations below we shall be concerned with a set A of integers. This is any given, finite or infinite, set of integers greater than 1 and subject to certain additional restrictions which will be stated later. The elements of A will be called a-numbers, and the letter a will be reserved for them. A number which is not divisible by any a-number will be called A-free, and our main concern will be with the study of A-free numbers. Their additive properties have recently been investigated elsewhere (Mirsky, 1948), and some estimates obtained in that investigation will be quoted in the present paper.

Further improvement on bounds for L-functions related to GL(3)

2019 ◽  
Vol 15 (07) ◽  
pp. 1487-1517 ◽  
Author(s):  
Haiwei Sun ◽  
Yangbo Ye
Keyword(s):  
Asymptotic Formula ◽  
Iterative Algorithm ◽  
Stationary Point ◽  
Kloosterman Sums ◽  
Maass Forms ◽  
Maass Form ◽  
New Techniques ◽  
Laplace Eigenvalue ◽  
Error Terms

Let [Formula: see text] be a fixed self-dual Hecke–Maass form for [Formula: see text], and let [Formula: see text] be an even Hecke–Maass form for [Formula: see text] with Laplace eigenvalue [Formula: see text], [Formula: see text]. A subconvexity bound for [Formula: see text] is improved to [Formula: see text], and a subconvexity bound for [Formula: see text] is improved to [Formula: see text]. New techniques employed include an application of an asymptotic formula by Salazar and Ye [Spectral square moments of a resonance sum for Maass forms, Front. Math. China 12(5) (2017) 1183–1200] to make error terms negligible, an iterative algorithm to locate stationary point, and a non-trivial estimation of Kloosterman sums.

scholarly journals Densities in certain three-way prime number races

2020 ◽  
pp. 1-34
Author(s):  
Jiawei Lin ◽  
Greg Martin
Keyword(s):  
Asymptotic Formula ◽  
Prime Number ◽  
Error Term ◽  
Proof Technique ◽  
Real Numbers ◽  
Logarithmic Density ◽  
Mutual Independence ◽  
Error Terms ◽  
The Relationship

Abstract Let $a_1$ , $a_2$ , and $a_3$ be distinct reduced residues modulo q satisfying the congruences $a_1^2 \equiv a_2^2 \equiv a_3^2 \ (\mathrm{mod}\ q)$ . We conditionally derive an asymptotic formula, with an error term that has a power savings in q, for the logarithmic density of the set of real numbers x for which $\pi (x;q,a_1)> \pi (x;q,a_2) > \pi (x;q,a_3)$ . The relationship among the $a_i$ allows us to normalize the error terms for the $\pi (x;q,a_i)$ in an atypical way that creates mutual independence among their distributions, and also allows for a proof technique that uses only elementary tools from probability.

scholarly journals Another generalisation of smith's determinant

1989 ◽  
Vol 40 (3) ◽  
pp. 413-415 ◽  
Author(s):  
Scott Beslin ◽  
Steve Ligh
Keyword(s):  
Greatest Common Divisor ◽  
Euler’S Totient Function ◽  
The Matrix ◽  
Positive Integers

Let S = {x1, x2, …, xn} be a set of distinct positive integers. The n × n matrix [S] = (Sij), where Sij, = (xi, xj), the greatest common divisor of xi, and xj, is called the greatest common divisor (GCD) matrix on S. H.J.S. Smith showed that the determinant of the matrix [E(n)], E(n) = { 1,2, …, n}, is ø(1)ø(2) … ø(n), where ø(x) is Euler's totient function. We extend Smith's result by considering sets S = {x1, x2, … xn} with the property that for all i and j, (xi, xj) is in S.

MORE ON A CERTAIN ARITHMETICAL DETERMINANT

2017 ◽  
Vol 97 (1) ◽  
pp. 15-25 ◽  
Author(s):  
ZONGBING LIN ◽  
SIAO HONG
Keyword(s):  
Linear Algebra ◽  
Acta Math ◽  
Arithmetical Function ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Mobius Function ◽  
Dirichlet Convolution ◽  
Positive Integers

Let $n\geq 1$ be an integer and $f$ be an arithmetical function. Let $S=\{x_{1},\ldots ,x_{n}\}$ be a set of $n$ distinct positive integers with the property that $d\in S$ if $x\in S$ and $d|x$. Then $\min (S)=1$. Let $(f(S))=(f(\gcd (x_{i},x_{j})))$ and $(f[S])=(f(\text{lcm}(x_{i},x_{j})))$ denote the $n\times n$ matrices whose $(i,j)$-entries are $f$ evaluated at the greatest common divisor of $x_{i}$ and $x_{j}$ and the least common multiple of $x_{i}$ and $x_{j}$, respectively. In 1875, Smith [‘On the value of a certain arithmetical determinant’, Proc. Lond. Math. Soc. 7 (1875–76), 208–212] showed that $\det (f(S))=\prod _{l=1}^{n}(f\ast \unicode[STIX]{x1D707})(x_{l})$, where $f\ast \unicode[STIX]{x1D707}$ is the Dirichlet convolution of $f$ and the Möbius function $\unicode[STIX]{x1D707}$. Bourque and Ligh [‘Matrices associated with classes of multiplicative functions’, Linear Algebra Appl. 216 (1995), 267–275] computed the determinant $\det (f[S])$ if $f$ is multiplicative and, Hong, Hu and Lin [‘On a certain arithmetical determinant’, Acta Math. Hungar. 150 (2016), 372–382] gave formulae for the determinants $\det (f(S\setminus \{1\}))$ and $\det (f[S\setminus \{1\}])$. In this paper, we evaluate the determinant $\det (f(S\setminus \{x_{t}\}))$ for any integer $t$ with $1\leq t\leq n$ and also the determinant $\det (f[S\setminus \{x_{t}\}])$ if $f$ is multiplicative.

scholarly journals Estimates of convolutions of certain number-theoretic error terms

2004 ◽  
Vol 2004 (1) ◽  
pp. 1-23 ◽  
Author(s):  
Aleksandar Ivic
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Abelian Groups ◽  
Divisor Problem ◽  
Dirichlet Divisor Problem ◽  
Error Terms ◽  
Convolution Function

Several estimates for the convolution functionC [f(x)]:=∫1xf(y) f(x/y)(dy/y)and its iterates are obtained whenf(x)is a suitable number-theoretic error term. We deal with the case of the asymptotic formula for∫0T|ζ(1/2+it)|2kdt(k=1,2), the general Dirichlet divisor problem, the problem of nonisomorphic Abelian groups of given order, and the Rankin-Selberg convolution.

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.

Sign in / Sign up

Export Citation Format

Share Document