The sixth, eighth, ninth, and tenth powers

of Ramanujan's theta function

Author:
Scott Ahlgren

Journal:
Proc. Amer. Math. Soc. **128** (2000), 1333-1338

MSC (1991):
Primary 11B65, 33D10

DOI:
https://doi.org/10.1090/S0002-9939-99-05181-3

Published electronically:
October 6, 1999

MathSciNet review:
1646322

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Abstract | References | Similar Articles | Additional Information

Abstract: In his Lost Notebook, Ramanujan claimed that the ``circular'' summation of the -th powers of the symmetric theta function satisfies a factorization of the form . Moreover, Ramanujan recorded identities expressing , , , , and in terms of his theta functions , , and . Ramanujan's claims were proved by Rangachari, and later (via elementary methods) by Son. In this paper we obtain similar identities for , , , and .

**[C-O]**H. Cohen and J. Oesterlé,*Dimensions des espaces de formes modulaires*, Springer Lect. Notes**627**(1976), 69-78. MR**57:12396****[G-H]**B. Gordon and K. Hughes,*Multiplicative properties of eta-products II*, Contemp. Math.**143**(1993), 415-430. MR**94a:11058****[K]**N. Koblitz,*Introduction to elliptic curves and modular forms*, Springer Verlag, New York, 1984. MR**86c:11040****[L]**G. Ligozat,*Courbes modulaires de genre 1*, Bull. Soc. Math. France**43**(1972), 1-80. MR**54:5121****[O]**K. Ono,*On the circular summation of the eleventh powers of Ramanujan's theta function*, J. Number Theory (to appear).**[Ram]**S. Ramanujan,*The lost notebook and other unpublished papers*, Narosa Publ. House, New Dehli, 1988. MR**89j:01078****[Ran]**S. Rangachari,*On a result of Ramanujan on theta functions*, J. Number Theory**48**(1994), 364-372. MR**95i:11038****[Sh]**G. Shimura,*On modular forms of half integral weight*, Annals of Math.**97**(1973), 440-481. MR**48:10989****[S]**S. Son,*Circular summations of theta functions in Ramanujan's lost notebook*, preprint.

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Additional Information

**Scott Ahlgren**

Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802-6401

Address at time of publication:
Department of Mathematics, Colgate University, Hamilton, New York 13346

Email:
ahlgren@math.psu.edu

DOI:
https://doi.org/10.1090/S0002-9939-99-05181-3

Keywords:
Ramanujan,
theta functions

Received by editor(s):
July 10, 1998

Published electronically:
October 6, 1999

Communicated by:
David E. Rohrlich

Article copyright:
© Copyright 2000
American Mathematical Society