Divisor function

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

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

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

[itex]\sigma_{1}(n)=\sum d[itex].

For example iff p is a prime number,

[itex]\sigma (p)=p+1\,\! [itex]

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

[itex]n = \prod_{i=1}^{r}p_{i}^{\alpha_{i}}[itex]

then we have

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

which is equivalent to the useful formula:

[itex]

\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}) [itex]

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

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

[itex]n = pq.[itex]

Then

[itex]\phi(n) = (p-1)(q-1) = n + 1 - (p+q),[itex]
[itex]\sigma(n) = (p+1)(q+1) = n + 1 + (p+q).[itex]
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Equations involving the divisor function

Two Dirichlet series involving the divisor function are:

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

and

[itex]\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)}[itex]

A Lambert series involving the divisor function is:

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

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:

[itex]

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

For the number of divisors function,

[itex]

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

Another bound on the number of divisors is

[itex]

\log d(n) < 1.066 \frac {\log n} {\log \log n} [itex] for n > 2.

For the sum of divisors function,

[itex]

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

Approximate growth rate

The growth rate of the sigma function is approximated by

[itex]

\sigma(n) \sim e^\gamma \ n\ \log \log n [itex]

where γ is Euler's constant.

References

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

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