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Divisor function

In mathematics the divisor function σ_a(n) is defined as the sum of the ath powers of the divisors of n, or

$\sigma_{a}(n)=\sum_{d|n} d^a\,\! .$

The notation d(n) is also used to denote σ₀(n), or the number of divisors of n. The sigma function σ(n) is

$\sigma_{1}(n)=\sum d$ .

For example iff p is a prime number,

$\sigma (p)=p+1\,\!$

because, by definition, the factors of a prime number are 1 and itself.

Generally, the divisor function is multiplicative, but not completely multiplicative.

The consequence of this is that, if we write

$n = \prod_{i=1}^{r}p_{i}^{\alpha_{i}}$

then we have

$\sigma(n) = \prod_{i=1}^{r} \frac{p_{i}^{\alpha_{i}+1}-1}{p_{i}-1}$

We also note $s (n) = σ(n) - n$ . This function is the one used to recognize perfect numbers which are the n for which $s (n) = n$ .

As an example, for two distinct primes p and q, let

n = p q .

Then

φ(n) = (p - 1)(q - 1) = n + 1 - (p + q),

σ(n) = (p + 1)(q + 1) = n + 1 + (p + q).

Two Dirichlet series involving the divisor function are:

$\sum_{n=1}^{\infty} \frac{\sigma_{a}(n)}{n^s}=\zeta(s) \zeta(s-a)$

and

$\sum_{n=1}^{\infty} \frac{\sigma_a(n)\sigma_b(n)}{n^s}=\frac{\zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)}{\zeta(2s-a-b)}$

A Lambert series involving the divisor function is:

$\sum_{n=1}^{\infty} q^n \sigma_a(n) = \sum_{n=1}^{\infty} \frac{n^a q^n}{1-q^n}$

for arbitrary complex |q| ≤ 1 and a. This summation also appears as the Fourier series of the Eisenstein series and the invariants of the Weierstrass elliptic functions.

Tom M.Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9

10-26-2009 08:16:03
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