{-# LANGUAGE ScopedTypeVariables #-}
module Math.NumberTheory.Zeta.Riemann
( zetas
, zetasEven
, zetasOdd
) where
import Data.ExactPi
import Data.Ratio ((%))
import Math.NumberTheory.Recurrences (bernoulli)
import Math.NumberTheory.Zeta.Hurwitz (zetaHurwitz)
import Math.NumberTheory.Zeta.Utils (intertwine, skipEvens, skipOdds)
zetasEven :: [ExactPi]
zetasEven :: [ExactPi]
zetasEven = (Integer -> Rational -> ExactPi)
-> [Integer] -> [Rational] -> [ExactPi]
forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith Integer -> Rational -> ExactPi
Exact [Integer
0, Integer
2 ..] ([Rational] -> [ExactPi]) -> [Rational] -> [ExactPi]
forall a b. (a -> b) -> a -> b
$ (Rational -> Rational -> Rational)
-> [Rational] -> [Rational] -> [Rational]
forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith Rational -> Rational -> Rational
forall a. Num a => a -> a -> a
(*) ([Rational] -> [Rational]
forall a. [a] -> [a]
skipOdds [Rational]
forall a. Integral a => [Ratio a]
bernoulli) [Rational]
cs
where
cs :: [Rational]
cs = (- Integer
1 Integer -> Integer -> Rational
forall a. Integral a => a -> a -> Ratio a
% Integer
2) Rational -> [Rational] -> [Rational]
forall a. a -> [a] -> [a]
: (Rational -> Integer -> Rational)
-> [Rational] -> [Integer] -> [Rational]
forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith (\Rational
i Integer
f -> Rational
i Rational -> Rational -> Rational
forall a. Num a => a -> a -> a
* (-Rational
4) Rational -> Rational -> Rational
forall a. Fractional a => a -> a -> a
/ Integer -> Rational
forall a. Num a => Integer -> a
fromInteger (Integer
2 Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
f Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* (Integer
2 Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
f Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
1))) [Rational]
cs [Integer
1..]
zetasOdd :: forall a. (Floating a, Ord a) => a -> [a]
zetasOdd :: a -> [a]
zetasOdd a
eps = (a
1 a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
0) a -> [a] -> [a]
forall a. a -> [a] -> [a]
: [a] -> [a]
forall a. [a] -> [a]
tail ([a] -> [a]
forall a. [a] -> [a]
skipEvens ([a] -> [a]) -> [a] -> [a]
forall a b. (a -> b) -> a -> b
$ a -> a -> [a]
forall a. (Floating a, Ord a) => a -> a -> [a]
zetaHurwitz a
eps a
1)
zetas :: (Floating a, Ord a) => a -> [a]
zetas :: a -> [a]
zetas a
eps = a
e a -> [a] -> [a]
forall a. a -> [a] -> [a]
: a
o a -> [a] -> [a]
forall a. a -> [a] -> [a]
: (a -> a -> a) -> [a] -> [a]
forall a. (a -> a -> a) -> [a] -> [a]
scanl1 a -> a -> a
forall a. (Ord a, Fractional a) => a -> a -> a
f ([a] -> [a] -> [a]
forall a. [a] -> [a] -> [a]
intertwine [a]
es [a]
os)
where
a
e : [a]
es = (ExactPi -> a) -> [ExactPi] -> [a]
forall a b. (a -> b) -> [a] -> [b]
map ((a -> a -> Bool) -> [Rational] -> a
forall a. Fractional a => (a -> a -> Bool) -> [Rational] -> a
getRationalLimit (\a
a a
b -> a -> a
forall a. Num a => a -> a
abs (a
a a -> a -> a
forall a. Num a => a -> a -> a
- a
b) a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
eps) ([Rational] -> a) -> (ExactPi -> [Rational]) -> ExactPi -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. ExactPi -> [Rational]
rationalApproximations) [ExactPi]
zetasEven
a
o : [a]
os = a -> [a]
forall a. (Floating a, Ord a) => a -> [a]
zetasOdd a
eps
f :: a -> a -> a
f a
x a
y = a
1 a -> a -> a
forall a. Ord a => a -> a -> a
`max` (a
y a -> a -> a
forall a. Ord a => a -> a -> a
`min` (a
1 a -> a -> a
forall a. Num a => a -> a -> a
+ (a
x a -> a -> a
forall a. Num a => a -> a -> a
- a
1) a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
2))