Jan Snellman: The ring of arithmetic functions under Dirichlet convolution; monomial ideals, Poincaré–Betti series, and a bit of AI

Abstract: An arithmetic function is a function defined on the positive integers; the name signals that number theory is about to happen. It is common to multiply two such functions using the Dirichlet convolution, which is the multiplication natural to Dirichlet series. It is also common to study the asymptotic growth of an arithmetic function f by looking at its values up to n.

What is less common is to study arithmetic functions truncated at n as an algebra; Johan Andersson suggested this problem to me sometime in the previous century, on the grounds that ”you get monomial algebras, the sort of thing you work with”.

On closer inspection they even turned out to be monomial algebras whose defining ideals are strongly stable, so that the homological behaviour (via Eliahou–Kervaire) is determined by simple combinatorial properties of the minimal monomial generators. As n grows one obtains a sequence of Betti numbers and so on, whose asymptotic growth is governed by the distribution of the primes in an intricate way.

I proved some simple properties of this in my paper, but recently I have had an AI look at the paper, and the AI

• found a known sieve function (Legendre's) that describes these algebras;
• proved a conjecture on the Poincaré–Betti series that I had left open in the paper;
• improved on my naive asymptotics by summing primes over various intervals, applying assorted inequalities, and generally tinkering and fiddling about like a genuine analytic number theorist.

I will describe my paper and the AI's contributions. If time permits, I will mention generalisations to convolutions other than Dirichlet's, such as unitary convolution.