View source for Sum of divisors

{{TEX|done}}{{MSC|11A25|11A51}}

''of a natural number $n$''

The sum of the positive integers divisors of a natural number $n$, including $1$ and $n$:
$$
\sigma(n) = \sum_{d | n} d \ .
$$
More generally, the function $\sigma_k$ is defined as
$$
\sigma_k(n) = \sum_{d | n} d^k \ .
$$
so that $\sigma = \sigma_1$ and the [[number of divisors]] function $\tau = \sigma_0$.

These are [[multiplicative arithmetic function]]s with [[Dirichlet series]]
$$
\sum_{n=1}^\infty \sigma_k(n) n^{-s} = \prod_p \left({(1-p^{-s})(1-p^{k-s}) }\right)^{-1} = \zeta(s) \zeta(s-k)\ .
$$

The average order of $\sigma(n)$ is given by
$$
\sum_{n \le x} \sigma(n) = \frac{\pi^2}{12} x^2 + O(x \log x) \ .
$$

There are a number of well-known classes of number characterised by their divisor sums.

A ''[[perfect number]]'' $n$ is the sum of its [[aliquot divisor]]s (those divisors other than $n$ itself), so $\sigma(n) = 2n$.  The even perfect numbers are characterised in terms of [[Mersenne prime]]s $P = 2^p-1$ as $n = 2^{p-1}.P$: it is not known if there are any odd perfect numbers.  An ''[[almost perfect number]]'' $n$ similarly has the property that $\sigma(n) = 2n-1$: these include the powers of 2.  A ''quasiperfect number'' is defined by $\sigma(n) = 2n+1$: it is not known if any exists.  See also [[Descartes number]].

====References====
* Kishore, Masao. "On odd perfect, quasiperfect, and odd almost perfect numbers". Mathematics of Computation '''36''' (1981) 583–586. ISSN 0025-5718.  {{ZBL|0472.10007}} 
* G. Tenenbaum, ''Introduction to Analytic and Probabilistic Number Theory'', Cambridge Studies in Advanced Mathematics '''46''', Cambridge University Press (1995) ISBN 0-521-41261-7 {{ZBL|0831.11001}}