An asymptotic expansion inspired by Ramanujan
This resolves a specific open problem in analytic number theory regarding Ramanujan's notebook, but is incremental in nature.
The paper proves that Ramanujan's asymptotic claim for a certain sum holds for n=2, correcting previous belief that it only holds for n=1, and discusses the error term and a correct generalization.
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.