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21: 25.2 Definition and Expansions

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25.2.11 $$\zeta \left(s\right)=\prod _p{(1-p^{-s})}^{-1},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $p$: prime number and $s$: complex variable

Keywords:

infinite product, primes

Source:

Apostol (1976, p. 231)

A&S Ref:

23.2.2

Referenced by:

§25.10(i)

Permalink:

http://dlmf.nist.gov/25.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iv), §25.2 and Ch.25

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22: Bibliography L

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List of Prime Numbers from 1 to 10,006,721. Publ. No. 165, Carnegie Institution of Washington, Washington, D.C..

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External Links:

ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

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23: Bibliography

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A Centennial History of the Prime Number Theorem. In Number Theory, Trends Math., pp. 1–14.

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External Links:

MathReview (Giovanni Coppola), ZentralBlatt

Referenced by:

(25.16.2), (25.16.4), §25.16(i), §25.16.

Encodings:

BibTeX

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24: 27.19 Methods of Computation: Factorization

… ►Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored. … ►As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard $p-1$, and a 67-digit prime for ecm. …

25: Bibliography E

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Generalized Bernoulli numbers, generalized irregular primes, and class number. Ann. Univ. Turku. Ser. A I 178, pp. 1–72.

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External Links:

ISSN 0082-7002, MathReview (Wells Johnson), ZentralBlatt

Referenced by:

§24.10(ii).

Encodings:

BibTeX

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26: Bibliography C

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Prime Numbers: A Computational Perspective. 2nd edition, Springer-Verlag, New York.

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External Links:

ISBN 978-0-387-25282-7; 0-387-25282-7, MathReview (Robert Juricevic), ZentralBlatt

Referenced by:

§27.12, §27.12, §27.18, §27.18, §27.18, §27.18, §27.18, §27.18, §27.19, §27.19, §27.19.

Encodings:

BibTeX

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27: 27.13 Functions

… ►Whereas multiplicative number theory is concerned with functions arising from prime factorization, additive number theory treats functions related to addition of integers. … ►Every even integer $n>4$ is the sum of two odd primes. In this case, $S(n)$ is the number of solutions of the equation $n=p+q$, where $p$ and $q$ are odd primes. …Vinogradov (1937) proves that every sufficiently large odd integer is the sum of three odd primes, and Chen (1966) shows that every sufficiently large even integer is the sum of a prime and a number with no more than two prime factors. …

28: 24.19 Methods of Computation

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§24.19(i) Bernoulli and Euler Numbers and Polynomials

►Equations (24.5.3) and (24.5.4) enable $B_n$ and $E_n$ to be computed by recurrence. …A similar method can be used for the Euler numbers based on (4.19.5). … ►

§24.19(ii) Values of $B_n$ Modulo $p$

… ►We list here three methods, arranged in increasing order of efficiency. …

29: Bibliography H

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On congruences involving Bernoulli numbers and irregular primes. II. Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.

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External Links:

ZentralBlatt

Referenced by:

§24.15(iii).

Encodings:

BibTeX

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30: Errata

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The largest known prime, which is a Mersenne prime, was updated from $2^{43,112,609}-1$ (2009) to $2^{82,589,933}-1$ (2018).

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