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A formula and a congruence for Ramanujan's $\tau$-function

Author(s): Matthew Papanikolas
Journal: Proc. Amer. Math. Soc. 134 (2006), 333-341.
MSC (2000): Primary 11F30; Secondary 11F33, 11T24, 33C99
Posted: June 14, 2005
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Abstract | References | Similar articles | Additional information

Abstract: We determine formulas for Ramanujan's $\tau$-function and for the coefficients of modular forms on $\Gamma_0(2)$ in terms of finite field ${}_3F_2$-hypergeometric functions. Using these formulas we obtain a new congruence of $\tau(p) \pmod{11}$.

References:

1.
S. Ahlgren and K. Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math. 518 (2000), 187-212. MR 1739404 (2001c:11057)

2.
S. Ahlgren, K. Ono, and D. Penniston, Zeta functions of an infinite family of $K3$ surfaces, Amer. J. Math. 124 (2002), 353-368. MR 1890996 (2003e:11068)

3.
S. Frechette, K. Ono, and M. Papanikolas, Gaussian hypergeometric functions and traces of Hecke operators, Int. Math. Res. Not. (2004), no. 60, 3233-3262. MR 2096220

4.
F. Q. Gouvêa, Non-ordinary primes: a story, Experiment. Math. 6 (1997), no. 3, 195-205. MR 1481589 (99a:11052)

5.
J. Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), 77-101. MR 0879564 (88e:11122)

6.
D. H. Lehmer, Ramanujan's function $\tau(n)$, Duke Math. J. 10 (1943), 483-492. MR 0008619 (5:35b)

7.
K. Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc. 350 (1998), 1205-1223. MR 1407498 (98e:11141)

8.
S. Ramanujan, On certain arithmetical functions, Trans. Camb. Phil. Soc. 22 (1916), 159-184.

9.
G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton University Press, Princeton, NJ, 1994, Reprint of the 1971 original. MR 1291394 (95e:11048)

10.
H. P. F. Swinnerton-Dyer, Congruence properties of $\tau(n)$, Ramanujan revisited (Urbana, IL, 1987), Academic Press, Boston, MA, 1988, pp. 289-311. MR 0938970 (89e:11028)

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Additional Information:

Matthew Papanikolas
Affiliation: Department of Mathematics, Texas A{&}M University, College Station, Texas 77843
Email: map@math.tamu.edu

DOI: 10.1090/S0002-9939-05-08029-9
PII: S 0002-9939(05)08029-9
Received by editor(s): April 27, 2004
Received by editor(s) in revised form: September 9, 2004
Posted: June 14, 2005
Additional Notes: This research was supported by NSF grant DMS-0340812 and NSA grant MDA904-03-1-0019
Communicated by: Wen-Ching Winnie Li
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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