Sum of divisors

2020 Mathematics Subject Classification: Primary: 11A25 Secondary: 11A51 [MSN][ZBL]

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 functions 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 divisors (those divisors other than $n$ itself), so $\sigma(n) = 2n$. The even perfect numbers are characterised in terms of Mersenne primes $P = 2^p-1$ 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