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