Math.NumberTheory.ArithmeticFunctions.Moebius

Values of Möbius function.

Documentation

data Moebius

Represents three possible values of Möbius function.

Constructors

MoebiusN

• 1

MoebiusZ

0

MoebiusP

1

Instances

Eq Moebius

Defined in Math.NumberTheory.ArithmeticFunctions.Moebius

Methods

(==) :: Moebius -> Moebius -> Bool

(/=) :: Moebius -> Moebius -> Bool

Ord Moebius

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

Defined in Math.NumberTheory.ArithmeticFunctions.Moebius

Methods

showsPrec :: Int -> Moebius -> ShowS

show :: Moebius -> String

showList :: [Moebius] -> ShowS

Semigroup Moebius

Defined in Math.NumberTheory.ArithmeticFunctions.Moebius

Methods

(<>) :: Moebius -> Moebius -> Moebius

sconcat :: NonEmpty Moebius -> Moebius

stimes :: Integral b => b -> Moebius -> Moebius

Monoid Moebius

Defined in Math.NumberTheory.ArithmeticFunctions.Moebius

Methods

mempty :: Moebius

mappend :: Moebius -> Moebius -> Moebius

mconcat :: [Moebius] -> Moebius

Unbox Moebius

Defined in Math.NumberTheory.ArithmeticFunctions.Moebius

Vector Vector Moebius

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

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

Defined in Math.NumberTheory.ArithmeticFunctions.Moebius

newtype Vector Moebius = V_Moebius (Vector Int8)

newtype MVector s Moebius

Defined in Math.NumberTheory.ArithmeticFunctions.Moebius

newtype MVector s Moebius = MV_Moebius (MVector s Int8)

runMoebius :: Num a => Moebius -> a

Convert to any numeric type.

sieveBlockMoebius :: Word -> Word -> Vector Moebius

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]