ASYMPTOTIC BEHAVIOR OF LARGEST EIGENVALUE OF MATRICES ASSOCIATED WITH COMPLETELY EVEN FUNCTIONS (MOD r)

2008 ◽  
Vol 01 (02) ◽  
pp. 225-235 ◽  
Author(s):  
Shaofang Hong
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bound ◽  
Multiplicative Function ◽  
Infinite Sequence ◽  
Arithmetical Function ◽  
Greatest Common Divisor ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Dirichlet Convolution ◽  
Positive Integers

Given an arbitrary strictly increasing infinite sequence [Formula: see text] of positive integers, let Sn = {x1,…, xn} for any integer n ≥ 1. Let q ≥ 1 be a given integer and f an arithmetical function. Let [Formula: see text] be the eigenvalues of the matrix (f(xi, xj)) having f evaluated at the greatest common divisor (xi, xj) of xi and xj as its i, j-entry. We obtain a lower bound depending only on x1 and n for [Formula: see text] if (f * μ)(d) < 0 whenever d|x for any x ∈ Sn, where g * μ is the Dirichlet convolution of f and μ. Consequently we show that for any sequence [Formula: see text], if f is a multiplicative function satisfying that f(pk) → ∞ as pk → ∞, f(2) > 1 and f(pm) ≥ f(2)f(pm−1) for any prime p and any integer m ≥ 1 and (f *μ)(d) > 0 whenever d|x for any [Formula: see text], then [Formula: see text] approaches infinity when n goes to infinity.

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 Some characterizations of totients

1996 ◽  
Vol 19 (2) ◽  
pp. 209-217 ◽  
Author(s):  
Pentti Haukkanen
Keyword(s):  
Multiplicative Function ◽  
Arithmetical Function ◽  
Multiplicative Functions ◽  
Dirichlet Convolution

An arithmetical function is said to be a totient if it is the Dirichlet convolution between a completely multiplicative function and the inverse of a completely multiplicative function. Euler's phi-function is a famous example of a totient. All completely multiplicative functions are also totients. There is a large number of characterizations of completely multiplicative functions in the literature, while characterizations of totients have not been widely studied in the literature. In this paper we present several arithmetical identities serving as characterizations of totients. We also introduce a new concrete example of a totient.

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.

Matrix Transformations Based on Dirichlet Convolution

1997 ◽  
Vol 40 (4) ◽  
pp. 498-508
Author(s):  
Chikkanna Selvaraj ◽  
Suguna Selvaraj
Keyword(s):  
Bounded Variation ◽  
Matrix Transformations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Summability Methods ◽  
The Matrix ◽  
Dirichlet Convolution ◽  
Necessary And Sufficient ◽  
Positive Integers

AbstractThis paper is a study of summability methods that are based on Dirichlet convolution. If f(n) is a function on positive integers and x is a sequence such that then x is said to be Af-summable to L. The necessary and sufficient condition for the matrix Af to preserve bounded variation of sequences is established. Also, the matrix Af is investigated as ℓ − ℓ and G − G mappings. The strength of the Af-matrix is also discussed.

scholarly journals On a subgroup of the group of multiplicative arithmetic functions

1975 ◽  
Vol 20 (3) ◽  
pp. 348-358 ◽  
Author(s):  
T. B. Carroll ◽  
A. A. Gioia
Keyword(s):  
Abelian Group ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Dirichlet Convolution ◽  
Image Position ◽  
Positive Integers ◽  
Multiplicative Arithmetic

An arithmetic function f is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n. Furthermore an arithmetic function is said to be linear (or completely multiplicative) if f(1) = 1 and f(mn) = f(m)f(n) for all positive integers m and n.The Dirichlet convolution of two arithmetic functions f and g is defined by for all n∈Z+. Recall that the set of all multiplicative functions, denoted by M, with this operation is an abelian group.

scholarly journals ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF RECIPROCAL POWER LCM MATRICES

2008 ◽  
Vol 50 (1) ◽  
pp. 163-174 ◽  
Author(s):  
SHAOFANG HONG ◽  
K. S. ENOCH LEE
Keyword(s):  
Asymptotic Behavior ◽  
Real Number ◽  
Infinite Sequence ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Lcm Matrix ◽  
Positive Integers ◽  
Power Lcm Matrix

AbstractLet$\{x_i\}_{i=1}^{\infty}$be an arbitrary strictly increasing infinite sequence of positive integers. For an integern≥1, let$S_n=\{x_1, {\ldots}\, x_n\}$. Letr>0 be a real number andq≥ 1 a given integer. Let$\lambda _n^{(1)}\, {\le}\, {\ldots}\, {\le}\, \lambda _n^{(n)}$be the eigenvalues of the reciprocal power LCM matrix$(\frac{1}{[x_i, x_j]^r})$having the reciprocal power${1\over {[x_i, x_j]^r}}$of the least common multiple ofxiandxjas itsi,j-entry. We show that the sequence$\{\lambda _n^{(q)}\}_{n=q}^{\infty}$converges and${\rm lim}_{n\, {\rightarrow}\, \infty}\lambda _n^{(q)}=0$. We show that the sequence$\{\lambda _n^{(n-q+1)}\}_{n=q}^{\infty}$converges if$s_r:=\sum_{i=1}^{\infty}{1\over {x_i^r}}<\infty $and${\rm lim}_{n\, {\rightarrow}\, \infty}\lambda _n^{(n-q+1)}\, {\le}\, s_r$. We show also that ifr> 1, then the sequence$\{\lambda _{ln}^{(tn-q+1)}\}_{n=1}^{\infty}$converges and${\rm lim}_{n\, {\rightarrow}\, \infty}\lambda _{ln}^{(tn-q+1)}=0$, wheretandlare given positive integers such thatt≤l−1.

scholarly journals Notes on the divisibility of GCD and LCM Matrices

2005 ◽  
Vol 2005 (6) ◽  
pp. 925-935 ◽  
Author(s):  
Pentti Haukkanen ◽  
Ismo Korkee
Keyword(s):  
Arithmetical Function ◽  
Greatest Common Divisor ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Positive Integers

LetS={x1,x2,…,xn}be a set of positive integers, and letfbe an arithmetical function. The matrices(S)f=[f(gcd(xi,xj))]and[S]f=[f(lcm [xi,xj])]are referred to as the greatest common divisor (GCD) and the least common multiple (LCM) matrices onSwith respect tof, respectively. In this paper, we assume that the elements of the matrices(S)fand[S]fare integers and study the divisibility of GCD and LCM matrices and their unitary analogues in the ringMn(ℤ)of then×nmatrices over the integers.

MASS PROBLEMS AND INITIAL SEGMENT COMPLEXITY

2014 ◽  
Vol 79 (01) ◽  
pp. 20-44 ◽  
Author(s):  
W. M. PHILLIP HUDELSON
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bound ◽  
Lower Bounds ◽  
Kolmogorov Complexity ◽  
Initial Segment ◽  
Infinite Sequence ◽  
Finite Sequence ◽  
Randomness Extraction ◽  
The Given ◽  
Infinite Sequences

Abstract By the complexity of a finite sequence of 0’s and 1’s we mean the Kolmogorov complexity, that is the length of the shortest input to a universal recursive function which returns the given sequence as output. By initial segment complexity of an infinite sequence of 0’s and 1’s we mean the asymptotic behavior of the complexity of its finite initial segments. In this paper, we construct infinite sequences of 0’s and 1’s with given recursive lower bounds on initial segment complexity which do not compute any infinite sequences of 0’s and 1’s with a significantly larger recursive lower bound on initial segment complexity. This improves several known results about randomness extraction and separates many natural degrees in the lattice of Muchnik degrees.

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]

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

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