On the Davison Convolution of Arithmetical Functions

1989 ◽  
Vol 32 (4) ◽  
pp. 467-473 ◽  
Author(s):  
Pentti Haukkanen
Keyword(s):  
Positive Integer ◽  
Arithmetical Functions ◽  
Positive Divisor ◽  
Complex Valued ◽  
Ordered Pairs

AbstractThe Davison convolution of arithmetical functions f and g is defined by where K is a complex-valued function on the set of all ordered pairs (n, d) such that n is a positive integer and d is a positive divisor of n. In this paper we shall consider the arithmetical equations f(r) = g, f(r) = fg, f o g = h in f and the congruence (f o g)(n) = 0 (mod n), where f(r) is the iterate of f with respect to the Davison convolution.

An Enumeration Problem Related to the Number of Labelled Bi-Coloured Graphs

1961 ◽  
Vol 13 ◽  
pp. 217-220 ◽  
Author(s):  
C. Y. Lee
Keyword(s):  
Positive Integer ◽  
Explicit Expression ◽  
Generating Function ◽  
Finite Sets ◽  
Enumeration Problem ◽  
Ordered Pairs

We will consider the following enumeration problem. Let A and B be finite sets with α and β elements in each set respectively. Let n be some positive integer such that n ≦ αβ. A subset S of the product set A × B of exactly n distinct ordered pairs (ai, bj) is said to be admissible if given any a ∈ A and b ∈ B, there exist elements (ai, bj) and (ak, bl) (they may be the same) in S such that ai = a and bl = b. We shall find here a generating function for the number N(α, β n) of distinct admissible subsets of A × B and from this generating function, an explicit expression for N(α, β n). In obtaining this result, the idea of a cut probability is used. This approach in a problem of enumeration may be of interest.

Quasi-uniqueness of the set of "Gaussian prime plus one's"

2014 ◽  
Vol 10 (07) ◽  
pp. 1783-1790
Author(s):  
Jay Mehta ◽  
G. K. Viswanadham
Keyword(s):  
Acta Arith ◽  
Gaussian Integers ◽  
Arithmetical Functions ◽  
Set Of Uniqueness ◽  
Positive Integers ◽  
Complex Valued ◽  
Gaussian Primes

We recall the well-known notion of the set of uniqueness for arithmetical functions, introduced by Kátai and several other mathematicians like Indlekofer, Elliot and Hoffman, independently. We define its analogue for completely additive complex-valued functions over the set of non-zero Gaussian integers with some examples. We show that the set of "Gaussian prime plus one's" along with finitely many Gaussian primes of norm up to some constant K is a set of uniqueness with respect to Gaussian integers. This is analogous to Kátai's result in the case of positive integers [I. Kátai, On sets characterizing number theoretical functions, II, Acta Arith.16 (1968) 1–14].

scholarly journals The Series Σ∞1 f(n)/n for Periodic f

1965 ◽  
Vol 8 (4) ◽  
pp. 413-432 ◽  
Author(s):  
Arthur E. Livingston
Keyword(s):  
Positive Integer ◽  
Complex Valued ◽  
Theoretic Function

We are here concerned with the problem of deciding when Σ∞n=1 f(n)/n ≠ 0, given that f is periodic and the series convergent. In particular, we considerConjecture A. Let p be a positive integer and f a (real-or complex-valued) number-theoretic function with period p.

scholarly journals Some New Notes on Mersenne Primes and Perfect Numbers

2020 ◽  
Vol 3 (1) ◽  
pp. 15
Author(s):  
Leomarich F Casinillo
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Mathematical Structure ◽  
Prime Numbers ◽  
Perfect Number ◽  
Positive Divisor ◽  
Triangular Numbers ◽  
Perfect Numbers ◽  
Mersenne Primes ◽  
Mersenne Prime

<p>Mersenne primes are specific type of prime numbers that can be derived using the formula <img title="\large M_p=2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&amp;space;M_p=2^{p}-1" alt="" />, where <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;space;p" alt="" /> is a prime number. A perfect number is a positive integer of the form <img title="\large P(p)=2^{p-1}(2^{p}-1)" src="https://latex.codecogs.com/gif.latex?\large&amp;space;P(p)=2^{p-1}(2^{p}-1)" alt="" /> where <img title="\large 2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&amp;space;2^{p}-1" alt="" /> is prime and <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;space;p" alt="" /> is a Mersenne prime, and that can be written as the sum of its proper divisor, that is, a number that is half the sum of all of its positive divisor. In this note, some concepts relating to Mersenne primes and perfect numbers were revisited. Further, Mersenne primes and perfect numbers were evaluated using triangular numbers. This note also discussed how to partition perfect numbers into odd cubes for odd prime <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;space;p" alt="" />. Also, the formula that partition perfect numbers in terms of its proper divisors were constructed and determine the number of primes in the partition and discuss some concepts. The results of this study is useful to better understand the mathematical structure of Mersenne primes and perfect numbers.</p>

scholarly journals Deficient perfect numbers with four distinct prime factors

2019 ◽  
Vol 13 (07) ◽  
pp. 2050126 ◽  
Author(s):  
Parama Dutta ◽  
Manjil P. Saikia
Keyword(s):  
Positive Integer ◽  
Perfect Number ◽  
Positive Divisor ◽  
Prime Factors ◽  
Perfect Numbers

For a positive integer [Formula: see text], if [Formula: see text] denotes the sum of the positive divisors of [Formula: see text], then [Formula: see text] is called a deficient perfect number if [Formula: see text] for some positive divisor [Formula: see text] of [Formula: see text]. In this paper, we prove some results about odd deficient perfect numbers with four distinct prime factors.

Relations between arithmetical functions

1967 ◽  
Vol 63 (4) ◽  
pp. 1027-1029
Author(s):  
C. J. A. Evelyn
Keyword(s):  
Positive Integer ◽  
Möbius Function ◽  
Natural Numbers ◽  
Arithmetical Functions ◽  
Mobius Function ◽  
Image Position ◽  
Fixed Positive Integer

In a recent note(1) I proved that if μ(n) denotes the usual Möbius function, N denotes a fixed positive integer and ifthenwhere T runs through all natural numbers ≤ x which are not divisible by an Nth power. In the present paper I shall establish some further relations of this character and, in particular, I shall prove that ifwherethenThus, in some respects, L(x) appears more regular than M(x), the sum over L(x/T) being multiplicative, whereas M(x1/N) is not.

Remainder sum and quotient sum function

2015 ◽  
Vol 07 (01) ◽  
pp. 1550001
Author(s):  
A. David Christopher
Keyword(s):  
Positive Integer ◽  
Periodic Function ◽  
Integer Partition ◽  
Integer Sequences ◽  
Partition Theory ◽  
Arithmetical Functions ◽  
Finite Set ◽  
Online Encyclopedia ◽  
Divisibility Properties ◽  
Positive Integers

This paper is concerned with two arithmetical functions namely remainder sum function and quotient sum function which are respectively the sequences A004125 and A006218 in Online Encyclopedia of Integer Sequences. The remainder sum function is defined by [Formula: see text] for every positive integer n, and quotient sum function is defined by [Formula: see text] where q(n, i) is the quotient obtained when n is divided by i. We establish few divisibility properties these functions enjoy and we found their bounds. Furthermore, we define restricted remainder sum function by RA(n) = ∑k∈A n mod k where A is a set of positive integers and we define restricted quotient sum function by QA(n) = ∑k∈A q(n, k). The function QA(n) is found to be a quasi-polynomial of degree one when A is a finite set of positive integers and RA(n) is found to be a periodic function with period ∏a∈A a. Finally, the above defined four functions found to have recurrence relation whose derivation requires few results from integer partition theory.

scholarly journals Properties of a Class of -Harmonic Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Elif Yaşar ◽  
Sibel Yalçın
Keyword(s):  
Differential Operator ◽  
Positive Integer ◽  
Harmonic Functions ◽  
Convex Combination ◽  
Extreme Points ◽  
Distortion Bounds ◽  
Harmonic Equation ◽  
Necessary And Sufficient ◽  
Complex Valued

A times continuously differentiable complex-valued function in a domain is -harmonic if satisfies the -harmonic equation , where is a positive integer. By using the generalized Salagean differential operator, we introduce a class of -harmonic functions and investigate necessary and sufficient coefficient conditions, distortion bounds, extreme points, and convex combination of the class.

scholarly journals An identity for a class of arithmetical functions of two variables

1988 ◽  
Vol 11 (2) ◽  
pp. 351-354
Author(s):  
J. Chidambaraswamy ◽  
P. V. Krishnaiah
Keyword(s):  
Positive Integer ◽  
Wide Class ◽  
Arithmetical Functions ◽  
Functions Of Two Variables ◽  
Special Cases ◽  
Ramanujan’S Sum ◽  
Ramanujan's Sum

For a positive integerr, letr∗denote the quotient ofrby its largest squarefree divisor(1∗=1). Recently, K. R. Johnson proved that(∗)∑d|n|C(d,r)|=r∗∏pa‖nr∗p+r(a+1)∏pa‖nr∗p|r(a(p−1)+1)   or   0according asr∗|nor not whereC(n,r)is the well known Ramanujan's sum. In this paper, using a different method, we generalize(∗)to a wide class of arithmetical functions of2variables and deduce as special cases(∗)and similar formulae for several generalizations of Ramanujan''s sum.

On a pair of zeta functions

2016 ◽  
Vol 12 (08) ◽  
pp. 2323-2342
Author(s):  
Zhi-Wei Sun
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Riemann Hypothesis ◽  
Zeta Functions ◽  
Prime Number Theorem ◽  
Arithmetical Functions ◽  
Prime Factors

Let [Formula: see text] be a positive integer, and define [Formula: see text] for [Formula: see text], where [Formula: see text] denotes the number of distinct prime factors of [Formula: see text], and [Formula: see text] represents the total number of prime factors of [Formula: see text] (counted with multiplicity). In this paper, we study these two zeta functions and related arithmetical functions. We show that [Formula: see text] which is similar to the known identity [Formula: see text] equivalent to the Prime Number Theorem. For [Formula: see text], we prove that [Formula: see text] We also raise a hypothesis on the parities of [Formula: see text] which implies the Riemann Hypothesis.

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