Divisor function

From Academic Kids

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

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

The notations d(n) and <math>\tau(n)<math> (the tau function) are also used to denote σ0(n), or the number of divisors of n. The sigma function σ(n) is

<math>\sigma_{1}(n)=\sum d<math>.

For example iff p is a prime number,

<math>\sigma (p)=p+1\,\! <math>

because, by definition, the factors of a prime number are 1 and itself. Clearly 1 < d(n) < n for all n > 1 and σ(n) > n for all n > 1.

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

The consequence of this is that, if we write

<math>n = \prod_{i=1}^{r}p_{i}^{\alpha_{i}}<math>

then we have

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

which is equivalent to the useful formula:

<math>

\sigma(n) = \prod_{i=1}^{r} \sum_{j=0}^{\alpha_{i}} p_{i}^{j} = \prod_{i=1}^{r} (1 + p_{i} + p_{i}^2 + ... + p_{i}^{a_r}) <math>

We also note <math>s(n) = \sigma(n) - n<math>. This function is the one used to recognize perfect numbers which are the n for which <math>s(n) = n<math>.

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

<math>n = pq.<math>

Then

<math>\phi(n) = (p-1)(q-1) = n + 1 - (p+q),<math>
<math>\sigma(n) = (p+1)(q+1) = n + 1 + (p+q).<math>
Contents

Equations involving the divisor function

Two Dirichlet series involving the divisor function are:

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

and

<math>\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)}<math>

A Lambert series involving the divisor function is:

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

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.

Inequalities with the divisor function

A pair of inequalities combining the divisor function and the φ function are:

<math>

\frac {6 n^2} {\pi^2} < \varphi(n) \sigma(n) < n^2 <math>, for n > 1.

For the number of divisors function,

<math>

d(n) < n^{\frac {2} {3}} <math> for n > 12.

Another bound on the number of divisors is

<math>

\log d(n) < 1.066 \frac {\log n} {\log \log n} <math> for n > 2.

For the sum of divisors function,

<math>

\sigma(n) < \frac {6n^\frac {3} {2}} {\pi^2} <math> for n > 12.

Approximate growth rate

The growth rate of the sigma function is approximated by

<math>

\sigma(n) \sim e^\gamma \ n\ \log \log n <math>

where γ is Euler's constant.

See also

References

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

ko:약수 함수

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