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

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 $$\gcd(a,b)=1$$.

• I'm not particularly knowledgeable about these phrases, but in general terminology can differ from author to author. I can certainly see someone calling such an $f$ "submultiplicative". "Weakly multiplicative" is not as descriptive in my mind, but I would not be even slightly surprised to find it used for the same condition. Apr 25 '20 at 17:55

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: $$\phi(0) = 1,\qquad \phi(x+y)\le \phi(x)\phi(y) \quad\text{for all x,y\in\mathbb{R}}.$$

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

• 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).