Copyright	(c) 2018 Andrew Lelechenko
License	MIT
Maintainer	Andrew Lelechenko <andrew.lelechenko@gmail.com>
Safe Haskell	None
Language	Haskell2010

Math.NumberTheory.ArithmeticFunctions.Moebius

Description

Values of Möbius function.

Synopsis

data Moebius
- = MoebiusN
- | MoebiusZ
- | MoebiusP
runMoebius :: Num a => Moebius -> a
sieveBlockMoebius :: Word -> Word -> Vector Moebius

Documentation

data Moebius Source #

Represents three possible values of Möbius function.

Constructors

MoebiusN	1
MoebiusZ	0
MoebiusP	1

Instances

Eq Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius Methods (==) :: Moebius -> Moebius -> Bool # (/=) :: Moebius -> Moebius -> Bool #
Ord Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius Methods compare :: Moebius -> Moebius -> Ordering # (<) :: Moebius -> Moebius -> Bool # (<=) :: Moebius -> Moebius -> Bool # (>) :: Moebius -> Moebius -> Bool # (>=) :: Moebius -> Moebius -> Bool # max :: Moebius -> Moebius -> Moebius # min :: Moebius -> Moebius -> Moebius #
Show Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius Methods showsPrec :: Int -> Moebius -> ShowS # show :: Moebius -> String # showList :: [Moebius] -> ShowS #
Semigroup Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius Methods (<>) :: Moebius -> Moebius -> Moebius # sconcat :: NonEmpty Moebius -> Moebius # stimes :: Integral b => b -> Moebius -> Moebius #
Monoid Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius Methods mempty :: Moebius # mappend :: Moebius -> Moebius -> Moebius # mconcat :: [Moebius] -> Moebius #
Unbox Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius
Vector Vector Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius Methods basicUnsafeFreeze :: PrimMonad m => Mutable Vector (PrimState m) Moebius -> m (Vector Moebius) # basicUnsafeThaw :: PrimMonad m => Vector Moebius -> m (Mutable Vector (PrimState m) Moebius) # basicLength :: Vector Moebius -> Int # basicUnsafeSlice :: Int -> Int -> Vector Moebius -> Vector Moebius # basicUnsafeIndexM :: Monad m => Vector Moebius -> Int -> m Moebius # basicUnsafeCopy :: PrimMonad m => Mutable Vector (PrimState m) Moebius -> Vector Moebius -> m () # elemseq :: Vector Moebius -> Moebius -> b -> b #
MVector MVector Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius Methods basicLength :: MVector s Moebius -> Int # basicUnsafeSlice :: Int -> Int -> MVector s Moebius -> MVector s Moebius # basicOverlaps :: MVector s Moebius -> MVector s Moebius -> Bool # basicUnsafeNew :: PrimMonad m => Int -> m (MVector (PrimState m) Moebius) # basicInitialize :: PrimMonad m => MVector (PrimState m) Moebius -> m () # basicUnsafeReplicate :: PrimMonad m => Int -> Moebius -> m (MVector (PrimState m) Moebius) # basicUnsafeRead :: PrimMonad m => MVector (PrimState m) Moebius -> Int -> m Moebius # basicUnsafeWrite :: PrimMonad m => MVector (PrimState m) Moebius -> Int -> Moebius -> m () # basicClear :: PrimMonad m => MVector (PrimState m) Moebius -> m () # basicSet :: PrimMonad m => MVector (PrimState m) Moebius -> Moebius -> m () # basicUnsafeCopy :: PrimMonad m => MVector (PrimState m) Moebius -> MVector (PrimState m) Moebius -> m () # basicUnsafeMove :: PrimMonad m => MVector (PrimState m) Moebius -> MVector (PrimState m) Moebius -> m () # basicUnsafeGrow :: PrimMonad m => MVector (PrimState m) Moebius -> Int -> m (MVector (PrimState m) Moebius) #
newtype Vector Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius newtype Vector Moebius = V_Moebius (Vector Int8)
newtype MVector s Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius newtype MVector s Moebius = MV_Moebius (MVector s Int8)

runMoebius :: Num a => Moebius -> a Source #

Convert to any numeric type.

sieveBlockMoebius :: Word -> Word -> Vector Moebius Source #

Evaluate the Möbius function over a block. Value of f at 0, if zero falls into block, is undefined.

Based on the sieving algorithm from p. 3 of Computations of the Mertens function and improved bounds on the Mertens conjecture by G. Hurst. It is approximately 5x faster than sieveBlockUnboxed.

>>> sieveBlockMoebius 1 10
[MoebiusP,MoebiusN,MoebiusN,MoebiusZ,MoebiusN,MoebiusP,MoebiusN,MoebiusZ,MoebiusZ,MoebiusP]