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Answering a question of Erdős, we show that a positive proportion of even numbers are in the form s(n), where s(n) = σ(n) - n, the sum of proper divisors of n.

The ring of arithmetical functions with unitary convolution: Divisorial and topological properties

Jan Snellman (2004)

Archivum Mathematicum

We study $(𝒜, +, \oplus)$ , the ring of arithmetical functions with unitary convolution, giving an isomorphism between $(𝒜, +, \oplus)$ and a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett [NumThe] between the ring $(𝒜, +, \cdot)$ of arithmetical functions with Dirichlet convolution and the power series ring $[[x_{1}, x_{2}, x_{3}, \dots]]$ on countably many variables. We topologize it with respect to a natural norm, and show that all ideals are quasi-finite. Some elementary results on factorization into atoms...

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Tau numbers: a partial proof of a conjecture and other results.

The average order of the Dirichlet series of the gcd-sum function.

The composition of the gcd and certain arithmetic functions.

The distribution of Farey numbers.

The distribution of totients.

The divisor function on residue classes I

The Fourier transform of functions of the greatest common divisor.

The general quasi-order algorithm in number theory.

The Generation of Arithmetical Identities.

The number of relatively prime subsets and phi functions for ${m, m + 1, \dots, n}$ .

The number of relatively prime subsets of ${1, 2, ..., n}$ .

The number of restricted solutions of some systems of linear congruences

The number of solutions of $ϕ (x) = m$ .

The range of the sum-of-proper-divisors function

The ring of arithmetical functions with unitary convolution: Divisorial and topological properties

The unitary amicable pairs to $10^{8}$ .

The unitary analogue of Pillai's arithmetical function.

Théorème concernant la fonction numérique $ρ_{2} (n)$ .

Théorèmes d'oscillations et fonction de Möbius.

There is no Odd Super Perfect Number of the Form p2... .

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