multiplicative number theory

SearchAdvancedHelp

(0.002 seconds)

11—20 of 21 matching pages

11: 13.27 Mathematical Applications

… ►where $\alpha$, $\beta$, $\gamma$, $\delta$ are real numbers, and $\gamma >0$. Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function. … ►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i). …

12: 27.5 Inversion Formulas

… ►The set of all number-theoretic functions $f$ with $f(1)\ne 0$ forms an abelian group under Dirichlet multiplication, with the function $\lfloor 1/n\rfloor$ in (27.2.5) as identity element; see Apostol (1976, p. 129). The multiplicative functions are a subgroup of this group. … ►Special cases of Möbius inversion pairs are: … ►Other types of Möbius inversion formulas include: … ►For a general theory of Möbius inversion with applications to combinatorial theory see Rota (1964). …

13: Bibliography M

… ►

The Theory of Groups. Clarendon Press, Oxford.

ⓘ

Notes:

Reprinted in 1988 by Krieger Publishing Co., Malabar, FL

External Links:

ISBN 0-19-853137-0;0-89464-287-1, MathReview (A. P. Street), ZentralBlatt

Referenced by:

§23.20(ii).

Encodings:

BibTeX

… ►

Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.

ⓘ

Notes:

A software package (the successor to CAYLEY) designed to solve computationally hard problems in algebra, number theory, geometry, and combinatorics, and to provide a mathematically rigorous environment for computing with algebraic, number-theoretic, combinatoric, and geometric objects.

External Links:

Magma

Referenced by:

4th item.

Encodings:

BibTeX

… ►

Zur Theorie des zylindrisch-parabolischen Spiegels. Z. Physik 118, pp. 343–356 (German).

ⓘ

External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§12.17.

Encodings:

BibTeX

… ►

Loud Speakers: Theory, Performance, Testing and Design. Oxford University Press, New York.

ⓘ

Notes:

Reprinted, with corrections, by Dover Publications Inc., 1960.

External Links:

MathReview, ZentralBlatt

Referenced by:

§11.12.

Encodings:

BibTeX

… ►

On the evaluation of some multiple series. J. London Math. Soc. (2) 33, pp. 368–371.

ⓘ

External Links:

Document, MathReview (V. F. Cowling), ZentralBlatt

Referenced by:

(25.6.5).

Encodings:

BibTeX

…

14: 27.14 Unrestricted Partitions

… ►Euler’s pentagonal number theorem states that …where the exponents $1$, $2$, $5$, $7$, $12$, $15$, $\mathrm{\dots }$ are the pentagonal numbers, defined by … ►For example, the Ramanujan identity … ►

§27.14(vi) Ramanujan’s Tau Function

… ►

15: Bibliography B

… ►

Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.

ⓘ

External Links:

Document, MathReview (T. M. Apostol), ZentralBlatt

Referenced by:

§24.16(ii).

Encodings:

BibTeX

►

Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.

ⓘ

External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§24.16(iii), §24.8(ii).

Encodings:

BibTeX

… ►

Efficiency and Security of Cryptosystems Based on Number Theory. Ph.D. Thesis, Swiss Federal Institute of Technology (ETH), Zurich.

ⓘ

External Links:

FullText

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

Pi and the AGM, A Study in Analytic Number Theory and Computational Complexity. Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York.

ⓘ

External Links:

ISBN 0-471-83138-7, MathReview (H. London), ZentralBlatt

Referenced by:

§19.35(i), §19.5, §19.8(i), §23.20(iv).

Encodings:

BibTeX

… ►

A Course in Computational Number Theory. Key College Publishing, Emeryville, CA.

ⓘ

Notes:

With 1 CD-ROM (Windows, Macintosh and UNIX) containing a collection of Mathematica programs for computations in number theory.

External Links:

ISBN 1-930190-10-7, MathReview (Samuel S. Wagstaff, Jr.), ZentralBlatt

Referenced by:

§27.12, §27.19, §27.21, 2nd item, Profile David M. Bressoud.

Encodings:

BibTeX

…

16: Bibliography V

… ►

Multiple Light Scattering. Vol. 1, Academic Press, New York.

ⓘ

Referenced by:

§6.17.

Encodings:

BibTeX

… ►

Some Wonderful Formulas $\mathrm{\dots }$ an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.

ⓘ

External Links:

MathReview (F. Beukers), ZentralBlatt

Referenced by:

(25.6.10), (25.6.8), (25.6.9), §25.6(i), §25.6.

Encodings:

BibTeX

… ►

Quantum Theory of Angular Momentum. World Scientific Publishing Co. Inc., Singapore.

ⓘ

External Links:

ISBN 9971-50-107-4, MathReview (M. A. Rashid), ZentralBlatt

Referenced by:

§16.24(iii), §34.1, §34.11, §34.13, §34.13, §34.14, §34.2, §34.3(iii), §34.3(vi), §34.4, §34.4, §34.5(iii), §34.5(vi), §34.7(ii), §34.7(iii), §34.8, §34.9, Ch.34.

Encodings:

BibTeX

… ►

Special Functions and the Theory of Group Representations. American Mathematical Society, Providence, RI.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§13.27.

Encodings:

BibTeX

… ►

The topological degree theory for the localization and computation of complex zeros of Bessel functions. Numer. Funct. Anal. Optim. 18 (1-2), pp. 227–234.

ⓘ

External Links:

ISSN 0163-0563, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

Encodings:

BibTeX

17: Bibliography K

… ►

Replica field theories, Painlevé transcendents, and exact correlation functions. Phys. Rev. Lett. 89 (25), pp. (250201–1)–(250201–4).

ⓘ

External Links:

ISSN 0031-9007, Document, MathReview, ZentralBlatt

Referenced by:

§32.16.

Encodings:

BibTeX

… ►

Remark on $p$-adic $q$-Bernoulli numbers. Adv. Stud. Contemp. Math. (Pusan) 1, pp. 127–136.

ⓘ

Notes:

Algebraic number theory (Hapcheon/Saga, 1996)

External Links:

ISSN 1229-3067, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

… ►

Neue Untersuchungen einer Reihe aus der Theorie der elliptischen Funktionen. Journal für die Reine und Angenwandte Mathematik 158, pp. 209–218 (German).

ⓘ

External Links:

ZentralBlatt

Referenced by:

§19.5.

Encodings:

BibTeX

… ►

Theorie und Anwendung der unendlichen Reihen. 4th edition, Die Grundlehren der mathematischen Wissenschaften, Band 2, Springer-Verlag, Berlin-Heidelberg (German).

ⓘ

Notes:

A translation by R. C. Young appeared in 1928, Theory and Application of Infinite Series, Blackie, 571pp.

External Links:

MathReview, ZentralBlatt

Referenced by:

§3.9(ii).

Encodings:

BibTeX

… ►

Zur Theorie der Modulformen $n$-ten Grades. I. Math. Z. 59, pp. 399–416 (German).

ⓘ

External Links:

ISSN 0025-5874, Document, MathReview (H. S. Zuckerman), ZentralBlatt

Referenced by:

§35.6(i).

Encodings:

BibTeX

…

18: Bibliography S

… ►

Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity. 4th edition, Springer-Verlag, Berlin.

ⓘ

Notes:

Volume 7 in Springer Series in Information Sciences.

External Links:

ZentralBlatt

Referenced by:

§27.16, §27.17.

Encodings:

BibTeX

… ►

Quantum defect theory. Rep. Prog. Phys. 46 (2), pp. 167–257.

ⓘ

External Links:

Document

Referenced by:

§33.14(i), §33.22(ii).

Encodings:

BibTeX

… ►

Efficient multiple-precision evaluation of elementary functions. Math. Comp. 52 (185), pp. 131–134.

ⓘ

External Links:

ISSN 0025-5718, FullText, MathReview (Menachem Dishon), ZentralBlatt

Referenced by:

§4.45(i).

Encodings:

BibTeX

… ►

Multiplicative processes and power laws. Phys. Rev. E 57 (4), pp. 4811–4813.

ⓘ

External Links:

Document

Referenced by:

§8.24(i).

Encodings:

BibTeX

… ►

Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Englewood Cliffs, N.J..

ⓘ

External Links:

ISBN 0130438936;978-0130438935, MathReview (R. D. Riess), ZentralBlatt

Referenced by:

§3.5(x).

Encodings:

BibTeX

…

19: Mathematical Introduction

… ►The NIST Handbook has essentially the same objective as the Handbook of Mathematical Functions that was issued in 1964 by the National Bureau of Standards as Number 55 in the NBS Applied Mathematics Series (AMS). … ►The first three chapters of the NIST Handbook and DLMF are methodology chapters that provide detailed coverage of, and references for, mathematical topics that are especially important in the theory, computation, and application of special functions. … ► ►►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
…
$\mathbb{N}$	set of all positive integers.
…
$n\mathbb{Z}$	set of all integer multiples of $n$.

… ►For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed. …

20: Errata

… ►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. … ►

17.11.2 $${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)=\frac{{(b,ax;q)}_{\mathrm{\infty }}}{{(c,x;q)}_{\mathrm{\infty }}}\sum _{n,r\geqq 0}\frac{{(a,b^{\prime };q)}_n{(c/b,x;q)}_r{b}^r{y}^n}{{(q,c^{\prime };q)}_n{(q;q)}_r{(ax;q)}_{n+r}}$$

The factor ${\left(q\right)}_r$ originally used in the denominator has been corrected to be ${(q;q)}_r$.

… ►

The multi-product notation ${(q,c;q)}_m{(q,c^{\prime };q)}_n$ in the denominator of the right-hand side was used.

… ►These additions were facilitated by an extension of the scheme for reference numbers; with “_” introducing intermediate numbers. … ►

17.2.22 $$\frac{{(q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}};q)}_n}{{(a^{\frac{1}{2}},-a^{\frac{1}{2}};q)}_n}=\frac{{(a{q}^2;q^2)}_n}{{(a;q^2)}_n}=\frac{1-a{q}^{2n}}{1-a}$$

17.2.23 $$\frac{{(q{a}^{\frac{1}{k}},q{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },q{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n}{{(a^{\frac{1}{k}},{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n}=\frac{{(a{q}^k;q^k)}_n}{{(a;q^k)}_n}=\frac{1-a{q}^{kn}}{1-a}$$

The numerators of the leftmost fractions were corrected to read ${(q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}};q)}_n$ and ${(q{a}^{\frac{1}{k}},q{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },q{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n$ instead of ${(q{a}^{\frac{1}{2}},-a{q}^{\frac{1}{2}};q)}_n$ and ${(a{q}^{\frac{1}{k}},q{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },q{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n$, respectively.

Reported 2017-06-26 by Jason Zhao.

…