# Ramanujan’s

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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 $\tau\left(n\right)$, and the values can be checked by the congruence (27.14.20). …
##### 2: 20.11 Generalizations and Analogs
###### §20.11(ii) Ramanujan’s Theta Function and $q$-Series
Ramanujans theta function $f(a,b)$ is defined by …
###### §20.11(iii) Ramanujan’s 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:
27.10.5 $c_{k}\left(n\right)=\sum_{d\mathbin{|}\left(n,k\right)}d\mu\left(\frac{k}{d}% \right).$
27.10.8 $a_{k}(m)=\sum_{d\mathbin{|}\left(m,k\right)}g(d)f\left(\frac{k}{d}\right)\frac% {d}{k}.$
Another generalization of Ramanujans sum is the Gauss sum $G\left(n,\chi\right)$ associated with a Dirichlet character $\chi\pmod{k}$. …
##### 5: 27.14 Unrestricted Partitions
###### §27.14(v) Divisibility Properties
For example, the Ramanujan identity …
###### §27.14(vi) Ramanujan’s Tau Function
The 24th power of $\eta\left(\tau\right)$ in (27.14.12) with $e^{2\pi\mathrm{i}\tau}=x$ is an infinite product that generates a power series in $x$ with integer coefficients called Ramanujans tau function $\tau\left(n\right)$: …
##### 6: 20.12 Mathematical Applications
For applications of Jacobi’s triple product (20.5.9) to Ramanujans $\tau\left(n\right)$ function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). …
##### 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.