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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
Published electronically: October 6, 1999
MathSciNet review: 1646322
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Abstract: In his Lost Notebook, Ramanujan claimed that the ``circular'' summation of the $n$-th powers of the symmetric theta function $f(a,b)$ satisfies a factorization of the form $f(a,b)F_{n}(ab)$. Moreover, Ramanujan recorded identities expressing $F_{2}(q)$, $F_{3}(q)$, $F_{4}(q)$, $F_{5}(q)$, and $F_{7}(q)$ in terms of his theta functions $\varphi (q)$, $\psi (q)$, and $f(-q)$. Ramanujan's claims were proved by Rangachari, and later (via elementary methods) by Son. In this paper we obtain similar identities for $F_{6}(q)$, $F_{8}(q)$, $F_{9}(q)$, and $F_{10}(q)$.

References [Enhancements On Off] (What's this?)

  • [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  • [Ram] S. Ramanujan, The lost notebook and other unpublished papers, Narosa Publ.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

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

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