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Ramanujan tau function

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1: 27.20 Methods of Computation: Other Number-Theoretic Functions
A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function τ ( n ) , and the values can be checked by the congruence (27.14.20). …
2: 27.21 Tables
Tables of the Ramanujan function τ ( n ) are published in Lehmer (1943) and Watson (1949). …
3: 27.14 Unrestricted Partitions
§27.14(vi) Ramanujan’s Tau Function
The 24th power of η ( τ ) in (27.14.12) with e 2 π i τ = x is an infinite product that generates a power series in x with integer coefficients called Ramanujan’s tau function τ ( n ) :
27.14.18 x n = 1 ( 1 - x n ) 24 = n = 1 τ ( n ) x n , | x | < 1 .
27.14.20 τ ( n ) σ 11 ( n ) ( mod 691 ) .
4: 20.12 Mathematical Applications
For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s τ ( n ) function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). …
5: Bibliography L
  • D. H. Lehmer (1943) Ramanujan’s function τ ( n ) . Duke Math. J. 10 (3), pp. 483–492.
  • D. H. Lehmer (1947) The vanishing of Ramanujan’s function τ ( n ) . Duke Math. J. 14 (2), pp. 429–433.
  • 6: Bibliography W
  • G. N. Watson (1949) A table of Ramanujan’s function τ ( n ) . Proc. London Math. Soc. (2) 51, pp. 1–13.
  • 7: Bibliography M
  • S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.
  • 8: 20.11 Generalizations and Analogs
    §20.11(ii) Ramanujan’s Theta Function and q -Series
    Ramanujan’s theta function f ( a , b ) is defined by …
    §20.11(iii) Ramanujan’s Change of Base
    However, in this case q is no longer regarded as an independent complex variable within the unit circle, because k is related to the variable τ = τ ( k ) of the theta functions via (20.9.2). … These results are called Ramanujan’s changes of base. …
    9: Bibliography K
  • S. L. Kalla (1992) On the evaluation of the Gauss hypergeometric function. C. R. Acad. Bulgare Sci. 45 (6), pp. 35–36.
  • K. S. Kölbig (1970) Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.
  • K. S. Kölbig (1972c) Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument. Comput. Phys. Comm. 4, pp. 221–226.
  • K. S. Kölbig (1981) A Program for Computing the Conical Functions of the First Kind P - 1 / 2 + i τ m ( x ) for m = 0 and m = 1 . Comput. Phys. Comm. 23 (1), pp. 51–61.
  • T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.