# divisor: Number theoretic functions In elliptic: Weierstrass and Jacobi Elliptic Functions

## Description

Various useful number theoretic functions

## Usage

 ```1 2 3 4 5 6``` ```divisor(n,k=1) primes(n) factorize(n) mobius(n) totient(n) liouville(n) ```

## Arguments

 `n,k` Integers

## Details

Functions `primes()` and `factorize()` cut-and-pasted from Bill Venables's conf.design package, version 0.0-3. Function `primes(n)` returns a vector of all primes not exceeding `n`; function `factorize(n)` returns an integer vector of nondecreasing primes whose product is `n`.

The others are multiplicative functions, defined in Hardy and Wright:

Function `divisor()`, also written sigma_k(n), is the divisor function defined on p239. This gives the sum of the k-th powers of all the divisors of `n`. Setting k=0 corresponds to d(n), which gives the number of divisors of `n`.

Function `mobius()` is the Moebius function (p234), giving zero if `n` has a repeated prime factor, and (-1)^q where n=p_1*p_2*...p_q otherwise.

Function `totient()` is Euler's totient function (p52), giving the number of integers smaller than `n` and relatively prime to it.

Function `liouville()` gives the Liouville function.

## Note

The divisor function crops up in `g2.fun()` and `g3.fun()`. Note that this function is not called `sigma()` to avoid conflicts with Weierstrass's sigma function (which ought to take priority in this context).

## Author(s)

Robin K. S. Hankin and Bill Venables (`primes()` and `factorize()`)

## References

G. H. Hardy and E. M. Wright, 1985. An introduction to the theory of numbers (fifth edition). Oxford University Press.

## Examples

 ```1 2 3 4 5 6 7 8 9``` ```mobius(1) mobius(2) divisor(140) divisor(140,3) plot(divisor(1:100,k=1),type="s",xlab="n",ylab="divisor(n,1)") plot(cumsum(liouville(1:1000)),type="l",main="does the function ever exceed zero?") ```

elliptic documentation built on May 2, 2019, 9:37 a.m.