Ramanujan%E2%80%99s

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11: 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)e^{2\pi\mathrm{i}mn/k},

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Defines:: $c_{\NVar{k}}\left(\NVar{n}\right)$ : Ramanujan’s sum
Symbols:: $\chi\left(\NVar{n}\right)$ : 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:: pMML, png, TeX
See also:: Annotations for §27.10 and Ch.27

… ►

27.10.5

c_{k}\left(n\right)=\sum_{d\mathbin{|}\left(n,k\right)}d\mu\left(\frac{k}{d}% \right).

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, $c_{\NVar{k}}\left(\NVar{n}\right)$ : Ramanujan’s sum, $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), $d$ : positive integer, $k$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.10.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §27.10 and Ch.27

… ►Another generalization of Ramanujan’s sum is the Gauss sum

G\left(n,\chi\right)

associated with a Dirichlet character

\chi\pmod{k}

. …In particular,

G\left(n,\chi_{1}\right)=c_{k}\left(n\right)

. …

12: Peter Paule

… ►Paule’s main research interests are computer algebra and algorithmic mathematics, together with connections to combinatorics, special functions, number theory, and other related fields. He is on the editorial boards for the Journal of Symbolic Computation and The Ramanujan Journal, and is Managing Editor of Annals of Combinatorics. … ►Paule was a member of the original editorial committee for the DLMF project, in existence from the mid-1990’s to the mid-2010’s. …

13: David M. Bressoud

… ►His books are Analytic and Combinatorial Generalizations of the Rogers-Ramanujan Identities, published in Memoirs of the American Mathematical Society 24, No. 227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special Relativity, published by Springer-Verlag in 1992, A Radical Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S. Wagon), published by Key College Press in 2000, and A Radical Approach to Lebesgue’s Theory of Integration, published by the Mathematical Association of America and Cambridge University Press in 2007. …

14: George E. Andrews

… ►He has a long-term interest in the work of S. Ramanujan, whose last notebook he unearthed in 1976. …

15: 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

… ► … ►These results are called Ramanujan’s changes of base. …

16: 19.35 Other Applications

… ►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute

\pi

to high precision (Borwein and Borwein (1987, p. 26)). ►

§19.35(ii) Physical

…

17: 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). …

18: 27.21 Tables

… ►7 of Abramowitz and Stegun (1964) also lists the factorizations in Glaisher’s Table I(a); Table 24. … ►Tables of the Ramanujan function

\tau\left(n\right)

are published in Lehmer (1943) and Watson (1949). …The bibliography in Lehmer (1941) gives references to the places in Dickson’s History where the older tables are cited. …

19: Frank Garvan

… ►He is managing editor of the Ramanujan Journal, a journal devoted to areas of mathematics influenced by Ramanujan. …

20: Bibliography

… ►

C. Adiga, B. C. Berndt, S. Bhargava, and G. N. Watson (1985) Chapter

16

of Ramanujan’s second notebook: Theta-functions and

q

-series. Mem. Amer. Math. Soc. 53 (315), pp. v+85.

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External Links:: ISSN 0065-9266, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §20.11(ii).
Encodings:: BibTeX

… ►

G. Almkvist and B. Berndt (1988) Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses,

\pi

, and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.

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External Links:: ISSN 0002-9890, FullText, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

… ►

D. E. Amos (1990) Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument. ACM Trans. Math. Software 16 (2), pp. 178–182.

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Notes:: Double- and single-precision Fortran, maximum accuracy 18S or 14S.
External Links:: ISSN 0098-3500, Document, GAMS (module 683; package TOMS), MathReview, ZentralBlatt
Referenced by:: 1st item, 1st item.
Encodings:: BibTeX

… ►

G. E. Andrews, R. A. Askey, B. C. Berndt, and R. A. Rankin (Eds.) (1988) Ramanujan Revisited. Academic Press Inc., Boston, MA.

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External Links:: ISBN 0-12-058560-X, ISBN 0-12-058560-X, MathReview, ZentralBlatt
Referenced by:: §20.11(ii), Profile Richard A. Askey.
Encodings:: BibTeX

… ►

G. E. Andrews (1984) Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.

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External Links:: ISSN 0030-8730, MathReview (David M. Bressoud), ZentralBlatt
Referenced by:: §17.12.
Encodings:: BibTeX

…