-- |
-- Module:      Math.NumberTheory.Zeta.Riemann
-- Copyright:   (c) 2016 Andrew Lelechenko
-- Licence:     MIT
-- Maintainer:  Andrew Lelechenko <andrew.lelechenko@gmail.com>
--
-- Riemann zeta-function.

{-# 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)

-- | Infinite sequence of exact values of Riemann zeta-function at even arguments, starting with @ζ(0)@.
-- Note that due to numerical errors conversion to 'Double' may return values below 1:
--
-- >>> approximateValue (zetasEven !! 25) :: Double
-- 0.9999999999999996
--
-- Use your favorite type for long-precision arithmetic. For instance, 'Data.Number.Fixed.Fixed' works fine:
--
-- >>> import Data.Number.Fixed
-- >>> approximateValue (zetasEven !! 25) :: Fixed Prec50
-- 1.00000000000000088817842111574532859293035196051773
--
zetasEven :: [ExactPi]
zetasEven :: [ExactPi]
zetasEven = forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith Integer -> Rational -> ExactPi
Exact [Integer
0, Integer
2 ..] forall a b. (a -> b) -> a -> b
$ forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith forall a. Num a => a -> a -> a
(*) (forall a. [a] -> [a]
skipOdds forall a. Integral a => [Ratio a]
bernoulli) [Rational]
cs
  where
    cs :: [Rational]
cs = (- Integer
1 forall a. Integral a => a -> a -> Ratio a
% Integer
2) forall a. a -> [a] -> [a]
: forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
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))) [Rational]
cs [Integer
1..]

-- | Infinite sequence of approximate values of Riemann zeta-function
-- at odd arguments, starting with @ζ(1)@.
zetasOdd :: forall a. (Floating a, Ord a) => a -> [a]
zetasOdd :: forall a. (Floating a, Ord a) => a -> [a]
zetasOdd a
eps = (a
1 forall a. Fractional a => a -> a -> a
/ a
0) forall a. a -> [a] -> [a]
: forall a. [a] -> [a]
tail (forall a. [a] -> [a]
skipEvens forall a b. (a -> b) -> a -> b
$ forall a. (Floating a, Ord a) => a -> a -> [a]
zetaHurwitz a
eps a
1)

-- | Infinite sequence of approximate (up to given precision)
-- values of Riemann zeta-function at integer arguments, starting with @ζ(0)@.
--
-- >>> take 5 (zetas 1e-14) :: [Double]
-- [-0.5,Infinity,1.6449340668482264,1.2020569031595942,1.0823232337111381]
--
-- Beware to force evaluation of @zetas !! 1@ if the type @a@ does not support infinite values
-- (for instance, 'Data.Number.Fixed.Fixed').
--
zetas :: (Floating a, Ord a) => a -> [a]
zetas :: forall a. (Floating a, Ord a) => a -> [a]
zetas a
eps = a
e forall a. a -> [a] -> [a]
: a
o forall a. a -> [a] -> [a]
: forall a. (a -> a -> a) -> [a] -> [a]
scanl1 forall {a}. (Ord a, Fractional a) => a -> a -> a
f (forall a. [a] -> [a] -> [a]
intertwine [a]
es [a]
os)
  where
    a
e : [a]
es = forall a b. (a -> b) -> [a] -> [b]
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) [ExactPi]
zetasEven
    a
o : [a]
os = forall a. (Floating a, Ord a) => a -> [a]
zetasOdd a
eps

    -- Cap-and-floor to improve numerical stability:
    -- 0 < zeta(n + 1) - 1 < (zeta(n) - 1) / 2
    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))