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 Ramanujan’s function $\tau \left(n\right)$, and the values can be checked by the congruence (27.14.20). …

2: 20.11 Generalizations and Analogs

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§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

… ►These results are called Ramanujan’s 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 Ramanujan’s sum: ►

27.10.4 $$c_k\left(n\right)=\sum _{m=1}^k{\chi }_1\left(m\right){\mathrm{e}}^{2\pi \mathrm{i}mn/k},$$

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Defines:

$c_k\left(n\right)$: Ramanujan’s sum

Symbols:

$\chi \left(n\right)$: Dirichlet character, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E4

Encodings:

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See also:

Annotations for §27.10 and Ch.27

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27.10.5 $$c_k\left(n\right)=\sum _{d|\left(n,k\right)}d\mu \left(\frac{k}{d}\right).$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $c_k\left(n\right)$: Ramanujan’s sum, $\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.10.E5

Encodings:

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See also:

Annotations for §27.10 and Ch.27

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27.10.8 $$a_k(m)=\sum _{d|\left(m,k\right)}g(d)f\left(\frac{k}{d}\right)\frac{d}{k}.$$

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Symbols:

$\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $k$: positive integer, $m$: positive integer, $f(n)$: function, $g(m)$: periodic function and $a_k(m)$: coefficients

Permalink:

http://dlmf.nist.gov/27.10.E8

Encodings:

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See also:

Annotations for §27.10 and Ch.27

►Another generalization of Ramanujan’s sum is the Gauss sum $G(n,\chi )$ associated with a Dirichlet character $\chi (modk)$. …

5: 27.14 Unrestricted Partitions

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§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 ${\mathrm{e}}^{2\pi \mathrm{i}\tau }=x$ is an infinite product that generates a power series in $x$ with integer coefficients called Ramanujan’s tau function $\tau \left(n\right)$: … ►

6: 20.12 Mathematical Applications

… ►For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s $\tau \left(n\right)$ function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). …

7: 5.13 Integrals

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Ramanujan’s Beta Integral

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8: 17.13 Integrals

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Ramanujan’s Integrals

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9: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

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Ramanujan’s ${}_1{}^{}\psi _1^{}$ Summation

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10: Bibliography R

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Congruence properties of partitions. Math. Z. 9 (1-2), pp. 147–153.

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External Links:

ISSN 0025-5874, ISSN 0025-5874, Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.14(v).

Encodings:

BibTeX

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Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,

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Referenced by:

§24.7(i).

Encodings:

BibTeX

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Collected Papers of Srinivasa Ramanujan. Chelsea Publishing Co., New York.

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Notes:

Reprinted with commentary by Bruce Berndt, AMS Chelsea Publishing Series, American Mathematical Society (2000).

External Links:

ISBN 0-8218-2076-1

Referenced by:

§8.11(v), G. H. Hardy and S. Ramanujan (1918).

Encodings:

BibTeX

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