| Copyright | (c) Andrew Lelechenko 2014-2020 |
|---|---|
| License | GPL-3 |
| Maintainer | andrew.lelechenko@gmail.com |
| Safe Haskell | None |
| Language | Haskell2010 |
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
- data TauabTheorem
- tauab :: Integer -> Integer -> (TauabTheorem, OptimizeResult)
- data TauabcTheorem
- tauabc :: Integer -> Integer -> Integer -> (TauabcTheorem, OptimizeResult)
- data TauabcdTheorem
- tauabcd :: Integer -> Integer -> Integer -> Integer -> (TauabcdTheorem, OptimizeResult)
- data Theorem
- data TauAResult
- tauA :: [Integer] -> TauAResult
Documentation
data TauabTheorem Source #
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
tauab :: Integer -> Integer -> (TauabTheorem, OptimizeResult) Source #
Compute Θ(a, b) for given a and b.
data TauabcTheorem Source #
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
| Eq TauabcTheorem Source # | |
Defined in Math.ExpPairs.Kratzel Methods (==) :: TauabcTheorem -> TauabcTheorem -> Bool # (/=) :: TauabcTheorem -> TauabcTheorem -> Bool # | |
| Ord TauabcTheorem Source # | |
Defined in Math.ExpPairs.Kratzel Methods compare :: TauabcTheorem -> TauabcTheorem -> Ordering # (<) :: TauabcTheorem -> TauabcTheorem -> Bool # (<=) :: TauabcTheorem -> TauabcTheorem -> Bool # (>) :: TauabcTheorem -> TauabcTheorem -> Bool # (>=) :: TauabcTheorem -> TauabcTheorem -> Bool # max :: TauabcTheorem -> TauabcTheorem -> TauabcTheorem # min :: TauabcTheorem -> TauabcTheorem -> TauabcTheorem # | |
| Show TauabcTheorem Source # | |
Defined in Math.ExpPairs.Kratzel Methods showsPrec :: Int -> TauabcTheorem -> ShowS # show :: TauabcTheorem -> String # showList :: [TauabcTheorem] -> ShowS # | |
| Pretty TauabcTheorem Source # | |
Defined in Math.ExpPairs.Kratzel | |
tauabc :: Integer -> Integer -> Integer -> (TauabcTheorem, OptimizeResult) Source #
Compute Θ(a, b, c) for given a, b and c.
data TauabcdTheorem Source #
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
| Eq TauabcdTheorem Source # | |
Defined in Math.ExpPairs.Kratzel Methods (==) :: TauabcdTheorem -> TauabcdTheorem -> Bool # (/=) :: TauabcdTheorem -> TauabcdTheorem -> Bool # | |
| Ord TauabcdTheorem Source # | |
Defined in Math.ExpPairs.Kratzel Methods compare :: TauabcdTheorem -> TauabcdTheorem -> Ordering # (<) :: TauabcdTheorem -> TauabcdTheorem -> Bool # (<=) :: TauabcdTheorem -> TauabcdTheorem -> Bool # (>) :: TauabcdTheorem -> TauabcdTheorem -> Bool # (>=) :: TauabcdTheorem -> TauabcdTheorem -> Bool # max :: TauabcdTheorem -> TauabcdTheorem -> TauabcdTheorem # min :: TauabcdTheorem -> TauabcdTheorem -> TauabcdTheorem # | |
| Show TauabcdTheorem Source # | |
Defined in Math.ExpPairs.Kratzel Methods showsPrec :: Int -> TauabcdTheorem -> ShowS # show :: TauabcdTheorem -> String # showList :: [TauabcdTheorem] -> ShowS # | |
| Pretty TauabcdTheorem Source # | |
Defined in Math.ExpPairs.Kratzel | |
tauabcd :: Integer -> Integer -> Integer -> Integer -> (TauabcdTheorem, OptimizeResult) Source #
Compute Θ(a, b, c, d) for given a, b, c and d.
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 |
data TauAResult Source #
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
| Eq TauAResult Source # | |
Defined in Math.ExpPairs.Kratzel | |
| Ord TauAResult Source # | |
Defined in Math.ExpPairs.Kratzel Methods compare :: TauAResult -> TauAResult -> Ordering # (<) :: TauAResult -> TauAResult -> Bool # (<=) :: TauAResult -> TauAResult -> Bool # (>) :: TauAResult -> TauAResult -> Bool # (>=) :: TauAResult -> TauAResult -> Bool # max :: TauAResult -> TauAResult -> TauAResult # min :: TauAResult -> TauAResult -> TauAResult # | |
| Show TauAResult Source # | |
Defined in Math.ExpPairs.Kratzel Methods showsPrec :: Int -> TauAResult -> ShowS # show :: TauAResult -> String # showList :: [TauAResult] -> ShowS # | |
| Pretty TauAResult Source # | |
Defined in Math.ExpPairs.Kratzel | |
tauA :: [Integer] -> TauAResult Source #
Compute Θ(a1, a2...) for given list [a1, a2...].