Math.ExpPairs.Kratzel

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).

Documentation

data TauabTheorem

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

Bounded TauabTheorem

Methods

minBound :: TauabTheorem

maxBound :: TauabTheorem

Enum TauabTheorem

Methods

succ :: TauabTheorem -> TauabTheorem

pred :: TauabTheorem -> TauabTheorem

toEnum :: Int -> TauabTheorem

fromEnum :: TauabTheorem -> Int

enumFrom :: TauabTheorem -> [TauabTheorem]

enumFromThen :: TauabTheorem -> TauabTheorem -> [TauabTheorem]

enumFromTo :: TauabTheorem -> TauabTheorem -> [TauabTheorem]

enumFromThenTo :: TauabTheorem -> TauabTheorem -> TauabTheorem -> [TauabTheorem]

Eq TauabTheorem

Methods

(==) :: TauabTheorem -> TauabTheorem -> Bool

(/=) :: TauabTheorem -> TauabTheorem -> Bool

Ord TauabTheorem

Methods

compare :: TauabTheorem -> TauabTheorem -> Ordering

(<) :: TauabTheorem -> TauabTheorem -> Bool

(<=) :: TauabTheorem -> TauabTheorem -> Bool

(>) :: TauabTheorem -> TauabTheorem -> Bool

(>=) :: TauabTheorem -> TauabTheorem -> Bool

max :: TauabTheorem -> TauabTheorem -> TauabTheorem

min :: TauabTheorem -> TauabTheorem -> TauabTheorem

Show TauabTheorem

Methods

showsPrec :: Int -> TauabTheorem -> ShowS

show :: TauabTheorem -> String

showList :: [TauabTheorem] -> ShowS

Pretty TauabTheorem

Methods

pretty :: TauabTheorem -> Doc

prettyList :: [TauabTheorem] -> Doc

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).

Instances

Eq TauabcTheorem

Methods

(==) :: TauabcTheorem -> TauabcTheorem -> Bool

(/=) :: TauabcTheorem -> TauabcTheorem -> Bool

Ord TauabcTheorem

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

Methods

showsPrec :: Int -> TauabcTheorem -> ShowS

show :: TauabcTheorem -> String

showList :: [TauabcTheorem] -> ShowS

Pretty TauabcTheorem

Methods

pretty :: TauabcTheorem -> Doc

prettyList :: [TauabcTheorem] -> Doc

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

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

data TauabcdTheorem

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

Constructors

HeathBrown

Tauabc TauabcTheorem

Kr611

Kr1992_2

Kr1992_31

Kr1992_32

Kr2010_1a

Kr2010_1b

Kr2010_2

Kr2010_3

CaoZhai

Instances

Eq TauabcdTheorem

Methods

(==) :: TauabcdTheorem -> TauabcdTheorem -> Bool

(/=) :: TauabcdTheorem -> TauabcdTheorem -> Bool

Ord TauabcdTheorem

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

Methods

showsPrec :: Int -> TauabcdTheorem -> ShowS

show :: TauabcdTheorem -> String

showList :: [TauabcdTheorem] -> ShowS

Pretty TauabcdTheorem

Methods

pretty :: TauabcdTheorem -> Doc

prettyList :: [TauabcdTheorem] -> Doc

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...)

Constructors

NoTheorem

Ivic

Ab TauabTheorem

Abc TauabcTheorem

Abcd TauabcdTheorem

Instances

Eq Theorem

Methods

(==) :: Theorem -> Theorem -> Bool

(/=) :: Theorem -> Theorem -> Bool

Ord Theorem

Methods

compare :: Theorem -> Theorem -> Ordering

(<) :: Theorem -> Theorem -> Bool

(<=) :: Theorem -> Theorem -> Bool

(>) :: Theorem -> Theorem -> Bool

(>=) :: Theorem -> Theorem -> Bool

max :: Theorem -> Theorem -> Theorem

min :: Theorem -> Theorem -> Theorem

Show Theorem

Methods

showsPrec :: Int -> Theorem -> ShowS

show :: Theorem -> String

showList :: [Theorem] -> ShowS

Pretty Theorem

Methods

pretty :: Theorem -> Doc

prettyList :: [Theorem] -> Doc

data TauAResult

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

Methods

(==) :: TauAResult -> TauAResult -> Bool

(/=) :: TauAResult -> TauAResult -> Bool

Ord TauAResult

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

Methods

showsPrec :: Int -> TauAResult -> ShowS

show :: TauAResult -> String

showList :: [TauAResult] -> ShowS

Pretty TauAResult

Methods

pretty :: TauAResult -> Doc

prettyList :: [TauAResult] -> Doc

tauA :: [Integer] -> TauAResult

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