Arithmetic mean is to addition as Harmonic mean is to ...?

Take $n$ real numbers $x_1,\ldots,x_n.$

The Arithmetic mean $A_n=\frac{1}{n}(x_1+\ldots+x_n)$ is the answer to the question: "Which number, when added up $n$ times, is equal to the sum of the $x_1,\ldots,x_n?$"

If the numbers are non-negative, then the Geometric mean $G_n=(x_1\ldots x_n)^{1/n}$ is the answer to the question: "Which number, when multiplied $n$ times by itself, is equal to the product of the $x_1,\ldots,x_n?$"

So the Arithmetic and Geometric and Geometric mean are associated with addition and multiplication, respectively.

My question:

Is there an operation, call it shmultiplication, such that the Harmonic mean $H_n=\frac{n}{\frac{1}{x_1}+\ldots+\frac{1}{x_n}}$ is the answer to the question: "Which number, when shmultiplied $n$ times with itself, is equal to the shmultiplication of the $x_1,\ldots,x_n?$" (And if yes, does shmultiplication have a proper name?)

• "shmultiplication" = addition of the reciprocals. Nov 9 '17 at 20:25
• @JackD'Aurizio the $n$ would be missing then, though, no? That is, we'd get $\frac{1}{\frac{1}{x_1}+\ldots+\frac{1}{x_n}}$ instead of $\frac{n}{\frac{1}{x_1}+\ldots+\frac{1}{x_n}}.$ Nov 9 '17 at 20:26
• $H_n$ is the standard notation for the $n$-th harmonic number, i.e. $\sum_{k=1}^{n}\frac{1}{n}$. The harmonic mean of $x_1,\ldots,x_n$ is usually denoted as $\text{HM}(x_1,\ldots,x_n)$ and it is the number $\alpha$ such that $\frac{1}{\alpha}+\ldots+\frac{1}{\alpha}=\frac{1}{x_1}+\ldots+\frac{1}{x_n}$. So the shmultiplication in your last paragraph is exactly the addition of the reciprocals (I am assuming $x_k>0$, of course). Nov 9 '17 at 20:29
• I misunderstood you at first, thank you. Nov 9 '17 at 20:31
• Analytic number theorists might have a name for it, considering $\frac{1}{\sum_{n=1}^\infty \frac{1}{n^s}}$ is shmultiplication that represents the probability a set of $s$ distinct integers are relatively prime. Series of that form are common and called Dirichlet generating functions. Nov 14 '20 at 12:06

I think the shmultiplication you’re looking for is

$$a\circ b = \frac{ab}{a+b} = (a^{-1} + b^{-1})^{-1}.$$

Associativity can easily be seen (at least when division by $0$ is avoided) by noting this is the conjugation of addition by taking inverses, so then the shmultiplication of $n$ values can be defined iteratively and is equivalent to the reciprocal of the sum of reciprocals.

Similarly, multiplication could be viewed as the conjugation of addition by taking logarithms/exponentials.

I’d want to call this a “harmonic sum”, but I can’t find any supporting evidence for this name.

I think the "schmultiplication" you are looking for is the binary operation $a \star b = \frac{1}{a} + \frac{1}{b}$, and I don't think it has a name. For example, take the case $n=2$. Then let $m = \frac{2}{\frac{1}{a} + \frac{1}{b}}$ (the harmonic mean of $a$ and $b$). Then if you "schmultiply" $m$ with itself twice you get $m \star m = \frac{2}{m} = \frac{1}{a} + \frac{1}{b}$ and this is exactly the "schmultiplication" of $a$ and $b$.

• What happens to $(a\star b)\star c$? Nov 9 '17 at 20:41
• Erick Wong: Good point. My answer is wrong. The binary operation I gave isn't even associative. Your answer looks correct. Nov 10 '17 at 21:28