Math.ExpPairs.Pair

Provides a set of initial exponent pairs, consisting of two points (0, 1), (1/2, 1/2) and a triangle with vertices in (1/6, 2/3), (2/13, 35/52) and (32/205, 269/410). The triangle is represented as a list of nodes of a net, covering the triangle.

Below A and B stands for van der Corput's processes. See Math.ExpPairs.Process for explanations.

Documentation

data Triangle

Vertices of the triangle of initial exponent pairs.

Constructors

Corput16

Usual van der Corput exponent pair (1/6, 2/3) = AB(0, 1).

HuxW87b1

An exponent pair (2/13, 35/52) from Huxley M. N. `Exponential sums and the Riemann zeta function' // Proceedings of the International Number Theory Conference held at Universite Laval in 1987, Walter de Gruyter, 1989, P. 417-423.

Hux05

An exponent pair (32/205, 269/410) from Huxley M. N. `Exponential sums and the Riemann zeta function V' // Proc. Lond. Math. Soc., 2005, Vol. 90, no. 1., P. 1--41.

Instances

Bounded Triangle

Enum Triangle

Eq Triangle

Ord Triangle

Show Triangle

data InitPair' t

Type to hold an initial exponent pair.

Constructors

Corput01

Usual van der Corput exponent pair (0, 1).

Corput12

Usual van der Corput exponent pair (1/2, 1/2) = B(0, 1).

Mix t t

Point from the interior of Triangle. Exactly Mix a b = a * Corput16 + b * HuxW87b1 + (1-a-b) * Hux05

Instances

Eq t => Eq (InitPair' t)

(Show t, Num t, Eq t) => Show (InitPair' t)

type InitPair = InitPair' Rational

Exponent pair built from rational fractions of Corput16, HuxW87b1 and Hux05

initPairs :: [InitPair]

The set of initial exponent pairs. It consists of Corput01, Corput12 and 496 = sum [1..31] Mix-points, which forms a uniform net over Triangle.

initPairToValue :: InitPair -> (Rational, Rational)

Convert initial exponent pair from its symbolic representation as InitPair to pair of rationals.