## 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.
## Comments

The generalisation mentioned in the abstract was
suggested by Brent and McMillan
[49].
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