On the Subrecursive Computability of Several Famous Constants

Abstract: For any class F of total functions in the set N of the natural numbers, we define the notion of F-computable real number. A real number α is called F- computable if there exist one-argument functions f, g and h in F such that for any n in N the distance between the rational number f(n) . g(n)over h(n)+1 and the number α is not greater than the reciprocal of n + 1. Most concrete real numbers playing a role in analysis can be easily shown to be ε3-computable (as usually, Em denotes the m-th Grzegorczyk class). Although (as it is proved in Section 5 of this paper) there exist ε3-computable real numbers that are not ε2-computable, we prove that π, e and other remarkable real numbers are ε2-computable (the number π proves to be even L-computable, where L is the class of Skolem’s lower elementary functions). However, only the rational numbers would turn out to be ε2-computable according to a definition of F-computability using 2n instead of n +1.