Ramanujan and Euler's constant

139. R. P. Brent, Ramanujan and Euler's constant, Proceedings of Symposia in Applied Mathematics, Vol. 48 (edited by W. Gautschi), American Mathematical Society, Providence, Rhode Island, 1994, 541-545. MR 95k:01022, MR~95j:00014. Presented at the American Mathematical Society "Mathematics of Computation 1943-1993" Meeting, Vancouver, August 1993.

A longer version appeared as Technical Report CMA-MR02-93/SMS-10-93 (arXiv:1004.5506v1) and as "An asymptotic expansion inspired by Ramanujan", Australian Mathematical Society Gazette 20 (1993), 149-155. MR 95b:33006.

Abstract: dvi (3K), pdf (81K), ps (29K).

Paper: dvi (11K), pdf (285K), ps (31K).

Technical Report: dvi (14K), pdf (164K), ps (68K).

Transparencies: dvi (8K), pdf (78K), ps (50K).

Abstract

Corollary 2, Entry 9, Chapter 4 of Ramanujan's first notebook claims that a certain sum (see the dvi or pdf version of the abstract) is asymptotic to ln x + gamma as x -> infinity, where x is a real variable in the sum and gamma is Euler's constant. Ramanujan's claim is known to be correct for the case n = 1, but incorrect for n > 2 (here n is an integer parameter in the sum). We show that the result is correct for n = 2. We also consider the order of the error term, and discuss a different, correct generalisation of the case n = 1.