prime number theorem

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11: 24.10 Arithmetic Properties

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§24.10(i) Von Staudt–Clausen Theorem

►Here and elsewhere in §24.10 the symbol $p$ denotes a prime number. …The denominator of $B_{2n}$ is the product of all these primes $p$. … ►Let $B_{2n}=N_{2n}/D_{2n}$, with $N_{2n}$ and $D_{2n}$ relatively prime and $D_{2n}>0$. … ►

§24.10(iv) Factors

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12: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_1$), (mod $m_2$), (mod $m_3$), and (mod $m_4$), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

13: 27.4 Euler Products and Dirichlet Series

… ►The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product. …In this case the infinite product on the right (extended over all primes $p$) is also absolutely convergent and is called the Euler product of the series. … ►Euler products are used to find series that generate many functions of multiplicative number theory. … ►

27.4.9 $$\sum _{n=1}^{\mathrm{\infty }}{2}^{\nu \left(n\right)}{n}^{-s}=\frac{{(\zeta \left(s\right))}^2}{\zeta \left(2s\right)},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\nu \left(n\right)$: number of distinct primes dividing a number, $\mathrm{\Re }$: real part and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.4.E9

Encodings:

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See also:

Annotations for §27.4 and Ch.27

… ►In (27.4.12) and (27.4.13) ${\zeta }^{\prime }\left(s\right)$ is the derivative of $\zeta \left(s\right)$.

14: 1.10 Functions of a Complex Variable

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Picard’s Theorem

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§1.10(iv) Residue Theorem

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Rouché’s Theorem

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Lagrange Inversion Theorem

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Extended Inversion Theorem

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15: 1.4 Calculus of One Variable

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Mean Value Theorem

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Fundamental Theorem of Calculus

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First Mean Value Theorem

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Second Mean Value Theorem

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§1.4(vi) Taylor’s Theorem for Real Variables

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16: 18.2 General Orthogonal Polynomials

… ► … ►Markov’s theorem states that … ►Under further conditions on the weight function there is an equiconvergence theorem, see Szegő (1975, Theorem 13.1.2). … ►Part of this theorem was already proved by Blumenthal (1898). … ►See Szegő (1975, Theorem 7.2). …

17: 27.8 Dirichlet Characters

§27.8 Dirichlet Characters

… ►An example is the principal character (mod $k$): … ►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 ${\left(\chi \left(n\right)\right)}^{\varphi \left(k\right)}=1$. …If $\left(n,k\right)=1$, then the characters satisfy the orthogonality relation … ►A divisor $d$ of $k$ is called an induced modulus for $\chi$ if …

18: Bibliography G

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Notes:

Information on Mersenne primes and the source code for the Great Internet Mersenne Prime Search.

External Links:

Home Page

Referenced by:

5th item.

Encodings:

BibTeX

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A theorem on the numerators of the Bernoulli numbers. Amer. Math. Monthly 97 (2), pp. 136–138.

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External Links:

ISSN 0002-9890, FullText, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.10(iv), §24.16(iii).

Encodings:

BibTeX

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Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.

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External Links:

ISSN 0026-9255, MathReview (Peter Stevenhagen), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Number-Divisor Tables. British Association Mathematical Tables, Vol. VIII, Cambridge University Press, Cambridge, England.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

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Multilateral summation theorems for ordinary and basic hypergeometric series in $\mathrm{U}(n)$ . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.

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External Links:

ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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19: 22.18 Mathematical Applications

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§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

… ►This provides an abelian group structure, and leads to important results in number theory, discussed in an elementary manner by Silverman and Tate (1992), and more fully by Koblitz (1993, Chapter 1, especially §1.7) and McKean and Moll (1999, Chapter 3). …With the identification $x=\mathrm{sn}(z,k)$, $y=d(\mathrm{sn}(z,k))/dz$, the addition law (22.18.8) is transformed into the addition theorem (22.8.1); see Akhiezer (1990, pp. 42, 45, 73–74) and McKean and Moll (1999, §§2.14, 2.16). The theory of elliptic functions brings together complex analysis, algebraic curves, number theory, and geometry: Lang (1987), Siegel (1988), and Serre (1973).

20: 23.20 Mathematical Applications

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§23.20(ii) Elliptic Curves

… ►The geometric nature of this construction is illustrated in McKean and Moll (1999, §2.14), Koblitz (1993, §§6, 7), and Silverman and Tate (1992, Chapter 1, §§3, 4): each of these references makes a connection with the addition theorem (23.10.1). … ► $K$ always has the form $T\times {\mathbb{Z}}^r$ (Mordell’s Theorem: Silverman and Tate (1992, Chapter 3, §5)); the determination of $r$, the rank of $K$, raises questions of great difficulty, many of which are still open. … ►

§23.20(v) Modular Functions and Number Theory

►For applications of modular functions to number theory see §27.14(iv) and Apostol (1990). …