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A Survey of Gcd-Sum Functions
TLDR
In this article, the authors survey properties of the gcd-sum function and its analogs and establish asymptotic formulae with remainder terms for the quadratic moment and the reciprocal of the GCS function.Abstract:
We survey properties of the gcd-sum function and of its analogs. As new results, we establish asymptotic formulae with remainder terms for the quadratic moment and the reciprocal of the gcd-sum function and for the function defined by the harmonic mean of the gcd’s.read more
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Menon's identity and arithmetical sums representing functions of several variables
TL;DR: In this article, the authors generalize Menon's identity by considering sums representing arithmetical functions of several variables, and give a formula for the number of cyclic subgroups of the direct product of several cyclic groups of arbitrary orders.
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Multiplicative Arithmetic Functions of Several Variables: A Survey
TL;DR: In this paper, general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution, are surveyed.
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The discrete Fourier transform of $r$-even functions
László Tóth,Pentti Haukkanen +1 more
TL;DR: In this paper, a detailed study of the discrete Fourier transform (DFT) of r-even arithmetic functions, which form a subspace of the space of R-periodic functions, is presented.
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On the average value of the least common multiple of k positive integers
Titus Hilberdink,László Tóth +1 more
TL;DR: In this article, an asymptotic formula with error term for the sum ∑ n 1, …, n k ≤ x f ( [ n 1, …, n k ] ), where x is the least common multiple of the positive integers, is given.
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On the subgroups of finite abelian groups of rank three
TL;DR: In this article, the subgroups of the group Zm Zn Zr were described and a simple formula for the total number s(m;n;r) of the subgroup, where m,n,r are arbitrary positive integers was derived.
References
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Book
An Introduction to the Theory of Numbers
TL;DR: The fifth edition of the introduction to the theory of numbers has been published by as discussed by the authors, and the main changes are in the notes at the end of each chapter, where the author seeks to provide up-to-date references for the reader who wishes to pursue a particular topic further and to present a reasonably accurate account of the present state of knowledge.
Book
The Theory of the Riemann Zeta-Function
TL;DR: The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it one of the most important tools in the study of prime numbers as mentioned in this paper.
Book
Multiplicative number theory
TL;DR: In this article, the General Modulus is used to describe the distribution of the Primes in arithmetic progression. But the explicit formula for psi(x,chi) is different from the explicit Formula for xi(s) and xi (s,chi).
Book
An introduction to the theory of numbers
Ivan Niven,H. S. Zuckerman +1 more
TL;DR: Divisibility congruence quadratic reciprocity and Quadratic forms some functions of number theory some diophantine equations Farey fractions and irrational numbers simple continued fractions primes and multiplicative number theory algebraic numbers the partition function the density of sequences of integers.
Book
History of the Theory of Numbers
Abstract: THE third and concluding volume of Prof. Dickson's great work deals first with the arithmetical. theory of binary quadratic forms. A long chapter on the class-number is contributed by Mr. G. H. Cresse. Next comes an account of existing knowledge on quadratic forms in three or more variables, followed by chapters on cubic forms, Hermitian and bilinear forms, and modular invariants and covariants.History of the Theory of Numbers.Prof. Leonard Eugene Dickson. Vol. 3: Quadratic and Higher Forms. With a Chapter on the Class Number by G. H. Cresse. (Publication No. 256.) Pp. v + 313. (Washington: Carnegie Institution, 1923.) 3.25 dollars.