Asymptotic and non-asymptotic results for a binary additive problem involving Piatetski-Shapiro numbers

For all α 1 , α 2 ∈ (1, 2) with 1/α 1 + 1/α 2 > 5/3, we show that the number of pairs (n 1 , n 2 ) of positive integers with, where Γ denotes the gamma function.Moreover, we show a non-asymptotic result for the same counting problem when α 1 , α 2 ∈ (1, 2) lie in a larger range than the above.Finally, we give some asymptotic formulas for similar counting problems in a heuristic way.