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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 Ramanujans function τ ( n ) , and the values can be checked by the congruence (27.14.20). …
2: 20.11 Generalizations and Analogs
§20.11(ii) Ramanujans Theta Function and q -Series
Ramanujans theta function f ( a , b ) is defined by …
§20.11(iii) Ramanujans Change of Base
These results are called Ramanujans changes of base. …
3: George E. Andrews
He has a long-term interest in the work of S. Ramanujan, whose last notebook he unearthed in 1976. …
4: 27.10 Periodic Number-Theoretic Functions
An example is Ramanujans sum:
27.10.4 c k ( n ) = m = 1 k χ 1 ( m ) e 2 π i m n / k ,
27.10.8 a k ( m ) = d | ( m , k ) g ( d ) f ( k d ) d k .
Another generalization of Ramanujans sum is the Gauss sum G ( n , χ ) associated with a Dirichlet character χ ( mod k ) . …
5: 27.14 Unrestricted Partitions
§27.14(v) Divisibility Properties
For example, the Ramanujan identity …
§27.14(vi) Ramanujans 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 Ramanujans tau function τ ( n ) : …
6: 20.12 Mathematical Applications
For applications of Jacobi’s triple product (20.5.9) to Ramanujans τ ( n ) function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). …
7: 17.13 Integrals
Ramanujans Integrals
8: Bibliography R
  • S. Ramanujan (1921) Congruence properties of partitions. Math. Z. 9 (1-2), pp. 147–153.
  • S. Ramanujan (1927) Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,
  • S. Ramanujan (1962) Collected Papers of Srinivasa Ramanujan. Chelsea Publishing Co., New York.
  • 9: 17.8 Special Cases of ψ r r Functions
    Ramanujans ψ 1 1 Summation
    10: 5.13 Integrals
    Ramanujans Beta Integral