exp-pairs-0.2.1.0: Linear programming over exponent pairs

Math.ExpPairs.Kratzel

Description

Asymmetric divisor problem

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

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

Constructors

 Kr511a Theorem 5.11, case a) Kr511b Theorem 5.11, case b) Kr512a Theorem 5.12, case a) Kr512b Theorem 5.12, case b)
Instances
 Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methods Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methods Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methods Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methods Source # Instance detailsDefined in Math.ExpPairs.Kratzel MethodsshowList :: [TauabTheorem] -> ShowS # Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methodspretty :: TauabTheorem -> Doc ann #prettyList :: [TauabTheorem] -> Doc ann #

Compute Θ(a, b) for given a and b.

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).
Instances
 Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methods Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methods Source # Instance detailsDefined in Math.ExpPairs.Kratzel MethodsshowList :: [TauabcTheorem] -> ShowS # Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methodspretty :: TauabcTheorem -> Doc ann #prettyList :: [TauabcTheorem] -> Doc ann #

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

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

Constructors

 HeathBrown Heath-Brown, 1978 Tauabc TauabcTheorem Kr611 Theorem 6.11 Kr1992_2 Krätzel, Estimates in the general divisor problem, Abh. Math. Sem. Univ. Hamburg 62 (1992), 191-206, Theorem 2 for p = 4 Kr1992_31 Ibidem, Theorem 3 for p = 4 under condition 3.1 Kr1992_32 Ibidem, Theorem 3 for p = 4 under condition 3.2 Kr2010_1a Kr2010_1b Kr2010_2 Kr2010_3 CaoZhai
Instances
 Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methods Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methods Source # Instance detailsDefined in Math.ExpPairs.Kratzel MethodsshowList :: [TauabcdTheorem] -> ShowS # Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methodspretty :: TauabcdTheorem -> Doc ann #prettyList :: [TauabcdTheorem] -> Doc ann #

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

data Theorem Source #

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

Constructors

 NoTheorem Ivic Ab TauabTheorem Abc TauabcTheorem Abcd TauabcdTheorem
Instances
 Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methods(==) :: Theorem -> Theorem -> Bool #(/=) :: Theorem -> Theorem -> Bool # Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methods(<) :: Theorem -> Theorem -> Bool #(<=) :: Theorem -> Theorem -> Bool #(>) :: Theorem -> Theorem -> Bool #(>=) :: Theorem -> Theorem -> Bool # Source # Instance detailsDefined in Math.ExpPairs.Kratzel MethodsshowList :: [Theorem] -> ShowS # Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methodspretty :: Theorem -> Doc ann #prettyList :: [Theorem] -> Doc ann #

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

Constructors

 Node Theorem OptimizeResult Combination TauAResult TauAResult Rational
Instances
 Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methods Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methods Source # Instance detailsDefined in Math.ExpPairs.Kratzel MethodsshowList :: [TauAResult] -> ShowS # Source # Instance detailsDefined in Math.ExpPairs.Kratzel Methodspretty :: TauAResult -> Doc ann #prettyList :: [TauAResult] -> Doc ann #

tauA :: [Integer] -> TauAResult Source #

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