{-| Module : Math.ExpPairs.MenzerNowak Copyright : (c) Andrew Lelechenko, 2014-2020 License : GPL-3 Maintainer : andrew.lelechenko@gmail.com Asymmetric divisor problem with congruence conditions Let τ_{a, b}(l_1, k_1; l_2, k_2; n) denote the number of integer (v, w) with v^a w^b = n, v ≡ l_1 (mod k_1), w ≡ l_2 (mod k_2). Menzer and Nowak (/Menzer H., Nowak W. G./ `On an asymmetric divisor problem with congruence conditions' \/\/ Manuscr. Math., 1989, Vol. 64, no. 1, P. 107-119) proved an asymptotic formula for Σ_{n ≤ x} τ_{a, b}(l_1, k_1; l_2, k_2; n) with an error term of order (x \/ k_1^a \/ k_2^b)^(Θ(a, b) + ε). They provided an expression for Θ(a, b) in terms of exponent pairs. -} module Math.ExpPairs.MenzerNowak ( menzerNowak ) where import Math.ExpPairs -- |Compute Θ(a, b) for given a and b. menzerNowak :: Integer -> Integer -> OptimizeResult menzerNowak a' b' = optimize [ K 1 + L 1 :/: K (a + b) + M (a + b) , K 1 :/: K (a + b) - L a + M a ] [] where a = fromInteger a' b = fromInteger b'