Where did the sum-of-divisors function come from?

Question

Doing a research project on a few number-theoretic functions, and I was curious, where does the sum-of-divisors function come from? Surely someone thought it up and made it a possibility. I'm talking about $\sigma(n)$ here. But, in the same context, where did the number of positive divisors function from as well, $\tau(n)$. Is there a mathematician who is famous for introducing these two functions?

If anyone knows of a reference that would lead me to the answer to this, I would be happy to take a look at it.

Thanks.

Answer 1

Key players regarding the first usage of the divisor function and the introduction of the divisor function symbol $\sigma$ were

• Leonhard Euler: De numeris amicabilibus, Opuscula varii argumenti, Vol. II (Berlin, 1750), p. 23-107; Commentationes arithmeticae, Vol. I (Petrograd, 1849), p102,103; *Opera Omnia (1st ser.), Vol. VI, p.21.

• Peter Barlow: Theory of Numbers in Encyclopaedia Metropolitana, Pure Sciences, Vol. I (1845), p. 648

• Allan Cunningham: Proc. London Math. Soc., Vol. XXXV (1902-3), p. 40

• L.E. Dickson: History of the Theory of Numbers, Vol. I (1919), p.42, 48, 53.

These references are cited in:

A History of Mathematical Notations, Vol. II, section: Signs in the theory of numbers, $\S 407$:

$\S 407$. Divisors of numbers; residues. - The notation $\int n$ for the sum of the divisors of $n$ was introduced by Euler; when $n$ is prime and $k$ an integer, he wrote

$$\int n^k=1+n+n^2+\ldots+n^k=\frac{n^{k+1}-1}{n-1}$$

Barlow employed the sign $\int$ to save repetition of the words divisible by, and the sign $\int\int$ to express of the form of.

Cunningham lets $\sigma(N)$ denote the sum of the sub-factors of $N$ (including $1$, but excluding $N$). It was found that, with most numbers, $\sigma^nN=1$, when the operation $(\sigma)$ is repeated often enough. Here $\sigma^2(n)$ means $\sigma\{\sigma(n)\}$.

Dickson writes $s(n)$ in place of Cunningham's $\sigma(n)$; Dickson lets $\sigma(n)$ represent the sum of the divisors (including $1$ and $n$) of $n$; he lets also $\sigma_k(n)$ represent the sum of the $k$th powers of the divisors of $n$.

Note: The symbols $\int$ and $\int\int$ which Barlow used were in fact only similar to the integral symbol and rotated counter-clockwise by $90$ degrees. I'm not able to write this symbol with \LaTex.

Answer 2

There is a famous Euler's article on the subject. I don't know any studies preceding it.

Answer 3

The sum-of-divisors functions seem quite natural to me, in the following way.

When dealing with power series $\displaystyle \sum_n a_n n^{-s}$ (or other generating functions, for that matter), multiplying two together gives

$$\sum_n a_n n^{-s} \cdot \sum_m b_m m^{-s} = \sum_n c_n n^{-s},$$

where

$$c_n = \sum_{d \mid n} a_d b_{n/d}.$$

This prompts one to better understand how these sums work, yielding the study of convolutions of functions $f$ and $g$ as $f \star g (n) = \displaystyle \sum_{d \mid n} f(d)g(n/d)$. As we frequently do, it's easiest to start with trivial cases. Setting $g(n) \equiv f(n) \equiv 1$ gives that $f\star g (n) = d(n)$, the number of divisors of $n$. If we only keep one function at $1$ and let the other be the identity function, like $f(n) \equiv 1$ and $g(n) = n$, then $f \star g (n) = \sigma_1(n)$ the sum of the divisors.

So certainly, by the time Dirichlet series were in common usage by the early 1800s, these functions arose naturally.