If an arithmetic function $f$ satisfies $f(mn) \leq f(m)f(n)$ (whenever $\gcd(m,n)=1$), is $f$ weakly multiplicative or submultiplicative?

Question

From the preprint On sums of the small divisors of a natural number (Lemma 1, page 2) by Douglas E. Iannucci:

We observe here that the function $a(n)$ is not multiplicative. It is, however, supermultiplicative:

• Lemma 1: If $m$ and $n$ are relatively prime natural numbers, then $a(mn) \geq a(m)a(n)$.

Here is my question:

If an arithmetic function $f$ satisfies $f(mn) \leq f(m)f(n)$ (whenever $\gcd(m,n)=1$), is $f$ weakly multiplicative or submultiplicative?

For example, the divisor sum $\sigma_1$, the abundancy index $I(x)=\sigma_1(x)/x$, and the deficiency function $D(x)=2x-\sigma_1(x)$ all satisfy $$\sigma_1(ab) \leq \sigma_1(a)\sigma_1(b)$$ $$I(ab) \leq I(a)I(b)$$ $$D(ab) \leq D(a)D(b),$$ where equality holds in the first two inequalities, and the last inequality holds, if $\gcd(a,b)=1$.

Answer 1

The term you are looking for is submultiplicative (or, sometimes, sub-multiplicative). This term is used in precisely the sense you describe throughout number theory and functional analysis.

The general idea is something like the following: given a group (or monoid, or semigroup, or ...) $G$ and an ordered group (or ...) $H$, a function $f : G \to H$ is submultiplicative if $f(ab) \le f(a)f(b)$ for all $a,b\in G$. In practice, $H$ is usually either the real numbers or the integers with multiplication, and $G$ is is typically one of the half-line $[0,\infty)$ with multiplication or $\mathbb{R}$ with addition. In number theory, $G$ may also be the integers or the natural numbers, with appropriate operations.

A couple of citations (taken from among the top hits on the Google Scholar search "submultiplicative" andd in no particular order):

• Submultiplicative moments of the supremum of a random walk with negative drift

  Let $\phi(x)$, $x \in \mathbb{R}$, be a submultiplicative function, i.e. $\phi(x)$ is a finite, positive, Borel measurable function with the following properties: $$ \phi(0) = 1,\qquad \phi(x+y)\le \phi(x)\phi(y) \quad\text{for all $x,y\in\mathbb{R}$}. $$

• A submultiplicative property of the psi function:

  We recall that a function $f:[0, \infty) \to \mathbb{N}$ is said to be submultiplicative on $[0, \infty)$, if $$f(xy)\le f(x)f(y) \qquad \text{for all $x\ge 0$ and $y\ge 0$}.$$

• Continuous convolution semigroups integrating a submultiplicative function:

  A Borel measurable function $\varphi$ of a locally compact group $G$ into the the interval $]0,\infty[$ is said to be submultiplicative if $\varphi(xy) \le \varphi(x)\varphi(y)$ for all $x,y\in G$ and if there exists a positive reaal number $c = c(\varphi)$ such that $\{x\in G: \varphi(x) \le c\}$ is a neighborhood of the identity $e$ of $G$.

The phrase "weakly multiplicative" also appears in the literature, but it generally means something quite different (or, at least, that is what a quick search seems to indicate—none of these papers on in fields where I regularly work).