| Copyright | (c) 2016 Andrew Lelechenko |
|---|---|
| License | MIT |
| Maintainer | Andrew Lelechenko <andrew.lelechenko@gmail.com> |
| Stability | Provisional |
| Portability | Non-portable (GHC extensions) |
| Safe Haskell | None |
| Language | Haskell2010 |
Math.NumberTheory.ArithmeticFunctions
Description
This module provides an interface for defining and manipulating arithmetic functions. It also defines several most widespreaded arithmetic functions.
Synopsis
- data ArithmeticFunction n a where
- ArithmeticFunction :: Monoid m => (Prime n -> Word -> m) -> (m -> a) -> ArithmeticFunction n a
- runFunction :: UniqueFactorisation n => ArithmeticFunction n a -> n -> a
- multiplicative :: Num a => (Prime n -> Word -> a) -> ArithmeticFunction n a
- divisors :: (UniqueFactorisation n, Num n, Ord n) => n -> Set n
- divisorsA :: forall n. (UniqueFactorisation n, Num n, Ord n) => ArithmeticFunction n (Set n)
- divisorsList :: (UniqueFactorisation n, Num n) => n -> [n]
- divisorsListA :: forall n. (UniqueFactorisation n, Num n) => ArithmeticFunction n [n]
- divisorsSmall :: (UniqueFactorisation n, Prime n ~ Prime Int) => n -> IntSet
- divisorsSmallA :: forall n. Prime n ~ Prime Int => ArithmeticFunction n IntSet
- tau :: (UniqueFactorisation n, Num a) => n -> a
- tauA :: Num a => ArithmeticFunction n a
- sigma :: (UniqueFactorisation n, Integral n) => Word -> n -> n
- sigmaA :: forall n. (UniqueFactorisation n, Integral n) => Word -> ArithmeticFunction n n
- totient :: (UniqueFactorisation n, Num n) => n -> n
- totientA :: forall n. (UniqueFactorisation n, Num n) => ArithmeticFunction n n
- jordan :: (UniqueFactorisation n, Num n) => Word -> n -> n
- jordanA :: forall n. (UniqueFactorisation n, Num n) => Word -> ArithmeticFunction n n
- ramanujan :: Integer -> Integer
- ramanujanA :: ArithmeticFunction Integer Integer
- moebius :: UniqueFactorisation n => n -> Moebius
- moebiusA :: ArithmeticFunction n Moebius
- data Moebius
- runMoebius :: Num a => Moebius -> a
- liouville :: (UniqueFactorisation n, Num a) => n -> a
- liouvilleA :: Num a => ArithmeticFunction n a
- additive :: Num a => (Prime n -> Word -> a) -> ArithmeticFunction n a
- smallOmega :: (UniqueFactorisation n, Num a) => n -> a
- smallOmegaA :: Num a => ArithmeticFunction n a
- bigOmega :: UniqueFactorisation n => n -> Word
- bigOmegaA :: ArithmeticFunction n Word
- carmichael :: (UniqueFactorisation n, Integral n) => n -> n
- carmichaelA :: forall n. (UniqueFactorisation n, Integral n) => ArithmeticFunction n n
- expMangoldt :: (UniqueFactorisation n, Num n) => n -> n
- expMangoldtA :: forall n. (UniqueFactorisation n, Num n) => ArithmeticFunction n n
Documentation
data ArithmeticFunction n a where #
A typical arithmetic function operates on the canonical factorisation of a number into prime's powers and consists of two rules. The first one determines the values of the function on the powers of primes. The second one determines how to combine these values into final result.
In the following definition the first argument is the function on prime's
powers, the monoid instance determines a rule of combination (typically
Product or Sum), and the second argument is convenient for unwrapping
(typically, getProduct or getSum).
Constructors
| ArithmeticFunction :: Monoid m => (Prime n -> Word -> m) -> (m -> a) -> ArithmeticFunction n a |
Instances
runFunction :: UniqueFactorisation n => ArithmeticFunction n a -> n -> a #
Convert to function. The value on 0 is undefined.
Multiplicative functions
multiplicative :: Num a => (Prime n -> Word -> a) -> ArithmeticFunction n a #
Create a multiplicative function from the function on prime's powers. See examples below.
divisorsA :: forall n. (UniqueFactorisation n, Num n, Ord n) => ArithmeticFunction n (Set n) #
The set of all (positive) divisors of an argument.
divisorsList :: (UniqueFactorisation n, Num n) => n -> [n] #
divisorsListA :: forall n. (UniqueFactorisation n, Num n) => ArithmeticFunction n [n] #
The unsorted list of all (positive) divisors of an argument, produced in lazy fashion.
divisorsSmall :: (UniqueFactorisation n, Prime n ~ Prime Int) => n -> IntSet #
divisorsSmallA :: forall n. Prime n ~ Prime Int => ArithmeticFunction n IntSet #
Same as divisors, but with better performance on cost of type restriction.
tau :: (UniqueFactorisation n, Num a) => n -> a #
tauA :: Num a => ArithmeticFunction n a #
The number of (positive) divisors of an argument.
tauA = multiplicative (\_ k -> k + 1)
sigma :: (UniqueFactorisation n, Integral n) => Word -> n -> n #
sigmaA :: forall n. (UniqueFactorisation n, Integral n) => Word -> ArithmeticFunction n n #
The sum of the k-th powers of (positive) divisors of an argument.
sigmaA = multiplicative (\p k -> sum $ map (p ^) [0..k]) sigmaA 0 = tauA
totient :: (UniqueFactorisation n, Num n) => n -> n #
totientA :: forall n. (UniqueFactorisation n, Num n) => ArithmeticFunction n n #
Calculates the totient of a positive number n, i.e.
the number of k with 1 <= k <= n and gcd n k == 1ℤ/(n).
jordan :: (UniqueFactorisation n, Num n) => Word -> n -> n #
jordanA :: forall n. (UniqueFactorisation n, Num n) => Word -> ArithmeticFunction n n #
Calculates the k-th Jordan function of an argument.
jordanA 1 = totientA
ramanujanA :: ArithmeticFunction Integer Integer #
Calculates the Ramanujan tau function
of a positive number n, using formulas given here
moebius :: UniqueFactorisation n => n -> Moebius #
moebiusA :: ArithmeticFunction n Moebius #
Calculates the Möbius function of an argument.
Represents three possible values of Möbius function.
Instances
runMoebius :: Num a => Moebius -> a #
Convert to any numeric type.
liouville :: (UniqueFactorisation n, Num a) => n -> a #
liouvilleA :: Num a => ArithmeticFunction n a #
Calculates the Liouville function of an argument.
Additive functions
additive :: Num a => (Prime n -> Word -> a) -> ArithmeticFunction n a #
Create an additive function from the function on prime's powers. See examples below.
smallOmega :: (UniqueFactorisation n, Num a) => n -> a #
smallOmegaA :: Num a => ArithmeticFunction n a #
Number of distinct prime factors.
smallOmegaA = additive (\_ _ -> 1)
bigOmega :: UniqueFactorisation n => n -> Word #
bigOmegaA :: ArithmeticFunction n Word #
Number of prime factors, counted with multiplicity.
bigOmegaA = additive (\_ k -> k)
Misc
carmichael :: (UniqueFactorisation n, Integral n) => n -> n #
carmichaelA :: forall n. (UniqueFactorisation n, Integral n) => ArithmeticFunction n n #
Calculates the Carmichael function for a positive integer, that is,
the (smallest) exponent of the group of units in ℤ/(n).
expMangoldt :: (UniqueFactorisation n, Num n) => n -> n #
expMangoldtA :: forall n. (UniqueFactorisation n, Num n) => ArithmeticFunction n n #
The exponent of von Mangoldt function. Use log expMangoldtA to recover von Mangoldt function itself.