CJO - Abstract - A PAGE ON EISENSTEIN SERIES IN RAMANUJAN'S LOST NOTEBOOK

Glasgow Mathematical Journal (2003), 45: 123-129 Cambridge University Press

Copyright © 2003 Glasgow Mathematical Journal Trust

doi:10.1017/S0017089502001076

Published online by Cambridge University Press 01 May 2003

A PAGE ON EISENSTEIN SERIES IN RAMANUJAN'S LOST NOTEBOOK

BRUCE C. BERNDT ^a1 ¹ and AE JA YEE ^a1 ²
^a1 Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA e-mail: berndt@math.uiuc.edu, yee@math.uiuc.edu

Abstract

Page 188 in Ramanujan's lost notebook is devoted to a certain class of infinite series connected with Euler's pentagonal number theorem. These series are represented in terms of Ramanujan's famous Eisenstein series $P, Q$, and $R$. The purpose of this paper is to prove all the formulas on page 188 and to show that one of them leads to an interesting, new recurrence formula for $\sigma (n)$, the sum of the positive divisors of $n$.

(Received October 23 2001)
(Accepted February 22 2002)

Maths Classification

11F11.

Dedication:
IN MEMORY OF ROBERT A. RANKIN

Footnotes

¹ Research partially supported by grant MDA904-00-1-0015 from the National Security Agency.

² Research partially supported by a grant from the Number Theory Foundation.

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