• On an approximate identity of Ramanujan

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/097/01-03/0313-0324

• # Keywords

Approximate identity; asymptotic behaviour; Rogers—Ramanujan identities; dilogarithm function

• # Abstract

In his second notebook, Ramanujan says that$$\frac{q}{{x + }}\frac{{q^4 }}{{x + }}\frac{{q^8 }}{{x + }}\frac{{q^{12} }}{{x + }} \cdots = 1 - \frac{{qx}}{{1 + }}\frac{{q^2 }}{{1 - }}\frac{{q^3 x}}{{1 + }}\frac{{q^4 }}{{1 - }} \cdots$$ “nearly” forq andx between 0 and 1. It is shown in what senses this is true. In particular, asq → 1 the difference between the left and right sides is approximately exp −c(x)/(l-q) wherec(x) is a function expressible in terms of the dilogarithm and which is monotone decreasing with c(0) = π2/4,c(1) = π2/5; thus the difference in question is less than 2· l0−85 forq = 0·99 and allx between 0 and 1.

• # Author Affiliations

D Zagier1 2

1. Max-Planck Institut für Mathematik, Gottfried-Claren-Straße 26, Bonn - D-5300, FRG
2. Department of Mathematics, University of Maryland, College Park, Maryland - 20742, USA

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 3
June 2019