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.

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Arithmetical functions of greatest common divisor. I

A Survey of Gcd-Sum Functions

We survey general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution. We introduce and investigate a new convolution, called gcd convolution. We define and study the convolutes of arithmetic functions of several variables, according to the different types of convolutions. We discuss the multiple Dirichlet series and Bell series and present certain arithmetic and asymptotic results of some special multiplicative functions arising from problems in number theory, group theory, and combinatorics. We give a new proof to obtain the asymptotic density of the set of ordered r-tuples of positive integers with pairwise relatively prime components and consider a similar question related to unitary divisors.

Multiplicative Arithmetic Functions of Several Variables: A Survey

The gcd-sum is an arithmetic function defined as the sum of the gcd’s of the first n integers with n : g(n) = P n=1 (i,n). The function arises in deriving asymptotic estimates for a lattice point counting problem. The function is multiplicative, and has polynomial growth. Its Dirichlet series has a compact representation in terms of the Riemann zeta function. Asymptotic forms for values of partial sums of the Dirichlet series at real values are derived, including estimates for error terms.

The gcd-sum function

We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always welcome.

background

Errata and Addenda to Mathematical Constants

On the average value of the least common multiple of k positive integers

https://www.ams.org/proc/1960-011-02/S0002-9939-1960-0111713-8/S0002-9939-1960-0111713-8.pdf

Arithmetical functions of greatest common divisor. I

Citations

A Survey of Gcd-Sum Functions

Multiplicative Arithmetic Functions of Several Variables: A Survey

The gcd-sum function

Errata and Addenda to Mathematical Constants

On the average value of the least common multiple of k positive integers

References

Die Analytische Zahlentheorie

Sur une formule utile dans la détermination de certaines valeurs asymptotiques

Le plus grand diviseur carré

Related Papers (5)

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A Survey of Gcd-Sum Functions

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Sur le plus grand commun diviseur de plusieurs nombres