Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

New representations of Ramanujan's tau function

Author(s): John A. Ewell
Journal: Proc. Amer. Math. Soc. 128 (2000), 723-726.
MSC (1991): Primary 11A25; Secondary 11B75
Posted: July 28, 1999
MathSciNet review: 1657735
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Abstract | References | Similar articles | Additional information

Abstract: Several formulas for Ramanujan's function $\tau$ , defined by

$\begin{displaymath}x \prod _1^{\infty}(1-x^n)^{24}=\displaystyle\sum _1^{\infty} \tau (n) x^n \quad (\vert x \vert < 1), \end{displaymath}$

are presented. We also present a congruence modulo 3 for some of the function values.

References:

1.: J. A. Ewell, Arithmetical consequences of a sextuple product identity, Rocky Mountain J. of Math., v. 25 (1995), 1287-1293. MR 97e:11129
2.: J. A. Ewell, A note on a Jacobian identity, Proc. Amer. Math. Soc., v. 126 (1998), 421-423. MR 98k:33030
3.: G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Fourth edition, Clarendon Press, Oxford, 1960.
4.: C.J. Moreno, A necessary and sufficient condition for the Riemann hypothesis for Ramanujan's zeta function, Illinois J. of Math., v. 18 (1974), 107-114. MR 48:8410
5.: S. Ramanujan, Collected papers, Chelsea, New York, 1962.
6.: R. Sivaramakrishnan, Classical theory of arithmetic functions, Dekker, New York, 1989. MR 90a:11001

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