An asymptotic expansion inspired by Ramanujan

TL;DR

This paper investigates Ramanujan's asymptotic claim for a sum involving a parameter n, confirming its accuracy for n=2, analyzing the error term, and proposing a corrected generalization for n=1.

Contribution

The paper provides a rigorous proof that Ramanujan's asymptotic formula holds for n=2 and offers a revised generalization for n=1, clarifying the limitations of the original claim.

Findings

Confirmed Ramanujan's claim for n=2

Analyzed the error term's order

Proposed a correct generalization for n=1

Abstract

Corollary 2, Entry 9, Chapter 4 of Ramanujan's first notebook claims that a certain sum is asymptotic to ln(x) + gamma, 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.