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ON η3(aτ)η3(bτ) WITH a+b=8

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 June 2008

HENG HUAT CHAN ,
SHAUN COOPER  and
WEN-CHIN LIAW
HENG HUAT CHAN*
Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, 117543, Singapore (email: matchh@nus.edu.sg)
SHAUN COOPER
Affiliation:
Albany Campus, Massey University, Private Bag 102 904, North Shore Mail Centre, Auckland, New Zealand (email: s.cooper@massey.ac.nz)
WEN-CHIN LIAW
Affiliation:
Department of Mathematics, National Chung Cheng University, Min-Hsiung, Chia-Yi, 62101, Taiwan, Republic of China (email: wcliaw@math.ccu.edu.tw)
*
✉For correspondence; e-mail: matchh@nus.edu.sg
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Abstract

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We prove an observation associated with η3(τ)η3(7τ) which is found on page 54 of Ramanujan’s Lost Notebook (S. Ramanujan, The Lost Notebook and Other Unpublished Papers (Narosa, New Delhi, 1988)). We then study functions of the type η3(aτ)η3(bτ) with a+b=8.

Keywords

spherical theta functionDirichlet seriesEuler productHecke operatormodular formeigenformLost NotebookRamanujan

MSC classification

Secondary: 11F03: Modular and automorphic functions 11F11: Holomorphic modular forms of integral weight 11F20: Dedekind eta function, Dedekind sums
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 84 , Issue 3 , June 2008 , pp. 301 - 313
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

The first author is partially supported by Academic Research Fund, National University of Singapore, R-146-000-103-112.

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