modular theorems

SearchAdvancedHelp

(0.001 seconds)

11—15 of 15 matching pages

11: 27.8 Dirichlet Characters

… ►For any character $\chi (modk)$, $\chi \left(n\right)\ne 0$ if and only if $\left(n,k\right)=1$, in which case the Euler–Fermat theorem (27.2.8) implies ${\left(\chi \left(n\right)\right)}^{\varphi \left(k\right)}=1$. … ►

27.8.6

ⓘ

Symbols:

$\chi \left(n\right)$: Dirichlet character, $\varphi \left(n\right)$: Euler’s totient, $\overline{z}$: complex conjugate, $\equiv$: modular equivalence, $k$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.8 and Ch.27

►A Dirichlet character $\chi (modk)$ is called primitive (mod $k$) if for every proper divisor $d$ of $k$ (that is, a divisor ), there exists an integer $a\equiv 1(modd)$, with $\left(a,k\right)=1$ and $\chi \left(a\right)\ne 1$. … ►

27.8.7 $$\chi \left(a\right)=1\text{ for all }a\equiv 1\text{ (mod }d\text{)},$$ $\left(a,k\right)=1$.

ⓘ

Symbols:

$\chi \left(n\right)$: Dirichlet character, $\left(m,n\right)$: greatest common divisor (gcd), $\equiv$: modular equivalence, $d$: positive integer and $k$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.8 and Ch.27

…

12: Bibliography R

… ►

Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.

ⓘ

External Links:

ISSN 0178-8051, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

… ►

13 Lectures on Fermat’s Last Theorem. Springer-Verlag, New York.

ⓘ

External Links:

ISBN 0-387-90432-8, MathReview (T. Metsänkylä), ZentralBlatt

Referenced by:

§24.10(iv), §24.17(iii).

Encodings:

BibTeX

… ►

Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.

ⓘ

External Links:

ISBN 978-1-107-15938-9, Document, Link, MathReview (Paul M. Jenkins), ZentralBlatt

Referenced by:

Profile Ranjan Roy.

Encodings:

BibTeX

… ►

Dawson’s integral and the sampling theorem. Computers in Physics 3 (2), pp. 85–87.

ⓘ

Referenced by:

§7.25(vi).

Encodings:

BibTeX

…

13: 27.12 Asymptotic Formulas: Primes

… ►

Prime Number Theorem

… ►For example, if $2^n\cancel{\equiv }2(modn)$, then $n$ is composite. … ►A Carmichael number is a composite number $n$ for which $b^n\equiv b(modn)$ for all $b\in \mathbb{N}$. …

14: 27.14 Unrestricted Partitions

… ►Euler’s pentagonal number theorem states that … ►

§27.14(iv) Relation to Modular Functions

►Dedekind sums occur in the transformation theory of the Dedekind modular function $\eta \left(\tau \right)$, defined by … ►For further properties of the function $\eta \left(\tau \right)$ see §§23.15–23.19. … ►implies $p\left(5n+4\right)\equiv 0(mod5)$. …

15: Bibliography S

… ►

Topics in Complex Function Theory. Vol. III: Abelian Functions and Modular Functions of Several Variables. Interscience Tracts in Pure and Applied Mathematics, No. 25, Wiley-Interscience, [John Wiley & Sons, Inc], New York-London-Sydney.

ⓘ

Notes:

Translated from the original German by E. Gottschling and M. Tretkoff. Reprinted by John Wiley & Sons, New York, 1989.

External Links:

ISBN 0-471-79090-7, MathReview, ZentralBlatt

Referenced by:

Ch.21.

Encodings:

BibTeX

… ►

Sturm oscillation and comparison theorems. In Sturm-Liouville theory, pp. 29–43.

ⓘ

External Links:

Document, Link, MathReview Entry

Referenced by:

§18.36(vi), §18.39(i).

Encodings:

BibTeX

►

Szegő’s Theorem and Its Descendants. Spectral Theory for $L^2$ Perturbations of Orthogonal Polynomials. M. B. Porter Lectures, Princeton University Press, Princeton, NJ.

ⓘ

External Links:

ISBN 978-0-691-14704-8, MathReview (Harry Dym)

Referenced by:

§18.2(xi).

Encodings:

BibTeX

…