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Divisor function

In mathematics the divisor function σa(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 = pq.

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.

See also

References

  • Tom M.Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9
Last updated: 10-29-2005 02:13:46