On one asymptotic formula for the Euler constant

Abstract

For efficient computation of γ, various approximations have been constructed (see [1–5]; for the only presently known fast computation method for the Euler constant, see [4, 5]). On the other hand, rational approximations for γ were constructed in order to prove its conjectural irrationality (see, e.g., [6, 7]). In the present paper we construct an asymptotic representation for γ using a sequence of rational numbers. In the construction we use the techniques from [5, 8]. The obtained asymptotic formula involves a partial sum of a divergent (semiconvergent; see [2, pp. 127 and 134]) series and therefore is probably useless (as well as (1)) for finding a good approximation for the Euler constant. To compute the Euler constant with any given accuracy, one should use approximations from [3–5]. Throughout the paper, we use the following notation: for a real x, the function fractional part of x is defined by y = {x} = x− [x], (2) where [x] is the integral part of x, i.e., an integer such that [x] ≤ x < [x] + 1;