Ramanujan%20change%20of%20base

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1: 20.11 Generalizations and Analogs

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20.11.1

G(m,n)=\sum\limits_{k=0}^{n-1}e^{-\pi ik^{2}m/n};

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Defines:: $G(m,n)$ : Gauss sum (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $m$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/20.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §20.11(i), §20.11 and Ch.20

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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. …

2: 20 Theta Functions

Chapter 20 Theta Functions

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3: 27.14 Unrestricted Partitions

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§27.14(v) Divisibility Properties

►Ramanujan (1921) gives identities that imply divisibility properties of the partition function. For example, the Ramanujan identity … ►

§27.14(vi) Ramanujan’s Tau Function

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4: Bibliography B

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G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:: Document
Referenced by:: §33.22(iv).
Encodings:: BibTeX

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R. J. Baxter (1981) Rogers-Ramanujan identities in the hard hexagon model. J. Statist. Phys. 26 (3), pp. 427–452.

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External Links:: ISSN 0022-4715, Document, MathReview, ZentralBlatt
Referenced by:: §17.17.
Encodings:: BibTeX

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B. C. Berndt, S. Bhargava, and F. G. Garvan (1995) Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.

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External Links:: ISSN 0002-9947, FullText, MathReview (J. Borwein), ZentralBlatt
Referenced by:: §15.8(v), §20.11(iii).
Encodings:: BibTeX

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B. C. Berndt (1989) Ramanujan’s Notebooks. Part II. Springer-Verlag, New York.

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External Links:: ISBN 0-387-96794-X, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §15.7.
Encodings:: BibTeX

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B. C. Berndt (1991) Ramanujan’s Notebooks. Part III. Springer-Verlag, Berlin-New York.

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External Links:: ISBN 0-387-97503-9, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §17.13.
Encodings:: BibTeX

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5: 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

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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)

. …

6: Bibliography K

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M. Katsurada (2003) Asymptotic expansions of certain

q

-series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.

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External Links:: ISSN 0065-1036, Document, Link, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §17.2(i).
Encodings:: BibTeX

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R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:: ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt
Referenced by:: §18.26(v), §18.26(v).
Encodings:: BibTeX

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10: Bibliography R

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S. Ramanujan (1921) 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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S. Ramanujan (1927) 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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S. Ramanujan (1962) 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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J. Raynal (1979) On the definition and properties of generalized

6

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

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B. Riemann (1899) Elliptische Functionen. Teubner, Leipzig.

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Notes:: Based on notes of lectures by B. Riemann edited and published by H. Stahl. Reprinted by UMI Books on Demand (Ann Arbor, MI, USA).
External Links:: ZentralBlatt
Referenced by:: §20.12(ii).
Encodings:: BibTeX

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Ramanujan%20change%20of%20base

1—10 of 491 matching pages

1: 20.11 Generalizations and Analogs

§20.11(ii) Ramanujan’s Theta Function and $q$ -Series

§20.11(iii) Ramanujan’s Change of Base

2: 20 Theta Functions

Chapter 20 Theta Functions

3: 27.14 Unrestricted Partitions

§27.14(v) Divisibility Properties

§27.14(vi) Ramanujan’s Tau Function

4: Bibliography B

5: 27.10 Periodic Number-Theoretic Functions

6: Bibliography K

7: 8 Incomplete Gamma and Related
Functions

8: 28 Mathieu Functions and Hill’s Equation

9: 23 Weierstrass Elliptic and Modular
Functions

10: Bibliography R

Ramanujan%20change%20of%20base

1—10 of 491 matching pages

1: 20.11 Generalizations and Analogs

§20.11(ii) Ramanujan’s Theta Function and q -Series

§20.11(iii) Ramanujan’s Change of Base

2: 20 Theta Functions

Chapter 20 Theta Functions

3: 27.14 Unrestricted Partitions

§27.14(v) Divisibility Properties

§27.14(vi) Ramanujan’s Tau Function

4: Bibliography B

5: 27.10 Periodic Number-Theoretic Functions

6: Bibliography K

7: 8 Incomplete Gamma and Related Functions

8: 28 Mathieu Functions and Hill’s Equation

9: 23 Weierstrass Elliptic and Modular Functions

10: Bibliography R

§20.11(ii) Ramanujan’s Theta Function and $q$ -Series

7: 8 Incomplete Gamma and Related
Functions

9: 23 Weierstrass Elliptic and Modular
Functions