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

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    $\begingroup$ 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. $\endgroup$ Apr 25 '20 at 17:55
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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):

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).

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