Ramanujan sum

SearchAdvancedHelp

(0.002 seconds)

1—10 of 21 matching pages

1: 27.10 Periodic Number-Theoretic Functions

… ►An example is Ramanujan’s sum: ►

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

ⓘ

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:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

… ►

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

ⓘ

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:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

… ►

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

ⓘ

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:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

… ►In particular, $G(n,{\chi }_1)=c_k\left(n\right)$. …

2: Bibliography B

… ►

A new method for investigating Euler sums. Ramanujan J. 4 (4), pp. 397–419.

ⓘ

External Links:

ISSN 1382-4090, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

(25.6.16), (25.6.17), (25.6.18), (25.6.19), (25.6.20), §25.6(iii), §25.6(iii), §25.6.

Encodings:

BibTeX

…

3: 27.14 Unrestricted Partitions

… ►

27.14.18 $$x\prod _{n=1}^{\mathrm{\infty }}{(1-x^n)}^{24}=\sum _{n=1}^{\mathrm{\infty }}\tau \left(n\right)x^n,$$ .

ⓘ

Defines:

$\tau \left(n\right)$: Ramanujan’s tau function

Symbols:

$n$: positive integer and $x$: real number

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.14.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(vi), §27.14 and Ch.27

… ►

27.14.19 $$\tau \left(m\right)\tau \left(n\right)=\sum _{d|\left(m,n\right)}{d}^{11}\tau \left(\frac{mn}{d^2}\right),$$ $m,n=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$\tau \left(n\right)$: Ramanujan’s tau function, $\left(m,n\right)$: greatest common divisor (gcd), $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.14.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(vi), §27.14 and Ch.27

… ►

27.14.20 $$\tau \left(n\right)\equiv {\sigma }_{11}\left(n\right)(mod691).$$

ⓘ

Symbols:

$\tau \left(n\right)$: Ramanujan’s tau function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\equiv$: modular equivalence and $n$: positive integer

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.14.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(vi), §27.14 and Ch.27

…

4: 20.11 Generalizations and Analogs

… ►

§20.11(i) Gauss Sum

… ►

§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. …

5: Bibliography M

… ►

Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

… ►

New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.

ⓘ

External Links:

ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

…

6: 20.12 Mathematical Applications

… ►For applications of ${\theta }_3(0,q)$ to problems involving sums of squares of integers see §27.13(iv), and for extensions see Estermann (1959), Serre (1973, pp. 106–109), Koblitz (1993, pp. 176–177), and McKean and Moll (1999, pp. 142–143). ►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: 17.2 Calculus

… ►

17.2.36 $$\sum _{j=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{j}\right){(-z)}^j={(1-z)}^n.$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: nonnegative integer, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.2.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(iii), §17.2 and Ch.17

… ►

17.2.45 $${\int }_0^{1}f(x)d_{q}x=(1-q)\sum _{j=0}^{\mathrm{\infty }}f(q^j)q^j,$$

ⓘ

Symbols:

$\int$: integral, $d_{q}x$: $q$-differential, $q$: complex base, $j$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.2.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(v), §17.2 and Ch.17

… ►

17.2.46 $${\int }_0^{a}f(x)d_{q}x=a(1-q)\sum _{j=0}^{\mathrm{\infty }}f(a{q}^j)q^j.$$

ⓘ

Symbols:

$\int$: integral, $d_{q}x$: $q$-differential, $q$: complex base, $j$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.2.E46

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(v), §17.2 and Ch.17

… ►provided that ${\sum }_{j=-\mathrm{\infty }}^{\mathrm{\infty }}f(q^j)q^j$ converges. ►

§17.2(vi) Rogers–Ramanujan Identities

…

8: 26.10 Integer Partitions: Other Restrictions

… ►where the last right-hand side is the sum over $m\ge 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. … ►where the inner sum is the sum of all positive odd divisors of $t$. … ►where the inner sum is the sum of all positive divisors of $t$ that are in $S$. ►

§26.10(iv) Identities

►Equations (26.10.13) and (26.10.14) are the Rogers–Ramanujan identities. …

9: 17.12 Bailey Pairs

… ►

17.12.1 $$\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n{\gamma }_n=\sum _{n=0}^{\mathrm{\infty }}{\beta }_n{\delta }_n,$$

ⓘ

Symbols:

$n$: nonnegative integer, ${\alpha }_n$: part of Bailey pair and ${\beta }_n$: part of Bailey pair

Permalink:

http://dlmf.nist.gov/17.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.12, §17.12 and Ch.17

… ►

${\beta }_n={\displaystyle \sum _{j=0}^n}{\alpha }_j{u}_{n-j}{v}_{n+j},$

►

${\gamma }_n={\displaystyle \sum _{j=n}^{\mathrm{\infty }}}{\delta }_j{u}_{j-n}{v}_{j+n}.$

… ► ►The Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is: …

10: Bibliography L

… ►

Ramanujan’s function $\tau (n)$ . Duke Math. J. 10 (3), pp. 483–492.

ⓘ

External Links:

Document, MathReview (H. S. Zuckerman), ZentralBlatt

Referenced by:

§27.20, §27.20, §27.21.

Encodings:

BibTeX

►

The vanishing of Ramanujan’s function $\tau (n)$ . Duke Math. J. 14 (2), pp. 429–433.

ⓘ

External Links:

Document, MathReview (R. A. Rankin), ZentralBlatt

Referenced by:

§27.14(vi).

Encodings:

BibTeX

… ►

A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Adv. in Math. 45 (1), pp. 21–72.

ⓘ

External Links:

ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.16.

Encodings:

BibTeX

►

Note sur la fonction $𝔎(w,x,s)={\sum }_{k=0}^{\mathrm{\infty }}\frac{e^{2k\pi ix}}{{(w+k)}^s}$ . Acta Math. 11 (1-4), pp. 19–24 (French).

ⓘ

External Links:

ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt

Referenced by:

§25.14(i), Notations K.

Encodings:

BibTeX

…