Copyright | (c) Andrew Lelechenko, 2014-2015 |
---|---|
License | GPL-3 |
Maintainer | andrew.lelechenko@gmail.com |
Stability | experimental |
Portability | POSIX |
Safe Haskell | None |
Language | Haskell2010 |
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).
- data TauabTheorem
- tauab :: Integer -> Integer -> (TauabTheorem, OptimizeResult)
- data TauabcTheorem
- tauabc :: Integer -> Integer -> Integer -> (TauabcTheorem, OptimizeResult)
Documentation
data TauabTheorem Source
Special type to specify the theorem of Krätzel1988, which provided the best estimate of Θ(a, b)
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)
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) Source
Compute Θ(a, b, c) for given a, b and c.