exp-pairs-0.1.5.2: Linear programming over exponent pairs

Copyright(c) Andrew Lelechenko 2014-2015
LicenseGPL-3
Maintainerandrew.lelechenko@gmail.com
Stabilityexperimental
PortabilityPOSIX
Safe HaskellNone
LanguageHaskell2010

Math.ExpPairs.Kratzel

Description

Let τ_{a, b}(n) denote the number of integer (v, w) with v^a w^b = n.

Let τ_{a, b, c}(n) denote the number of integer (v, w, z) with v^a w^b z^c = n.

Krätzel (Krätzel E. `Lattice points'. Dordrecht: Kluwer, 1988) proved asymptotic formulas for Σ_{n ≤ x} τ_{a, b}(n) with an error term of order x^(Θ(a, b) + ε) and for Σ_{n ≤ x} τ_{a, b, c}(n) with an error term of order x^(Θ(a, b, c) + ε). He also provided a set of theorems to estimate Θ(a, b) and Θ(a, b, c).

Synopsis

Documentation

tauab :: Integer -> Integer -> (TauabTheorem, OptimizeResult) #

Compute Θ(a, b) for given a and b.

data TauabcTheorem #

Special type to specify the theorem of Krätzel1988, which provided the best estimate of Θ(a, b, c)

Constructors

Kolesnik

Kolesnik (Kolesnik G. `On the estimation of multiple exponential sums' // Recent progress in analytic number theory, London: Academic Press, 1981, Vol. 1, P. 231–246) proved that Θ(1, 1, 1) = 43 /96.

Kr61

Theorem 6.1

Kr62

Theorem 6.2

Kr63

Theorem 6.3

Kr64

Theorem 6.4

Kr65

Theorem 6.5

Kr66

Theorem 6.6

Tauab TauabTheorem

In certain cases Θ(a, b, c) = Θ(a, b).

tauabc :: Integer -> Integer -> Integer -> (TauabcTheorem, OptimizeResult) #

Compute Θ(a, b, c) for given a, b and c.

tauabcd :: Integer -> Integer -> Integer -> Integer -> (TauabcdTheorem, OptimizeResult) #

Compute Θ(a, b, c, d) for given a, b, c and d.

data Theorem #

Special type to specify the theorem of Krätzel1988, which provided the best estimate of Θ(a1, a2...)

data TauAResult #

Special type to specify the theorem of Krätzel1988, which provided the best estimate of Θ(a1, a2...)

tauA :: [Integer] -> TauAResult #

Compute Θ(a1, a2...) for given list [a1, a2...].