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21: 27.12 Asymptotic Formulas: Primes

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27.12.2 $$p_n>n\ln n,$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$\ln z$: principal branch of logarithm function, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.12.E2

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12 and Ch.27

… ►where $\lambda (\alpha )$ depends only on $\alpha$, and $\varphi \left(m\right)$ is the Euler totient function (§27.2). … ►For current records see The Great Internet Mersenne Prime Search. ►A pseudoprime test is a test that correctly identifies most composite numbers. …Descriptions and comparisons of pseudoprime tests are given in Bressoud and Wagon (2000, §§2.4, 4.2, and 8.2) and Crandall and Pomerance (2005, §§3.4–3.6). …

22: Bibliography K

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On the evaluation of the Gauss hypergeometric function. C. R. Acad. Bulgare Sci. 45 (6), pp. 35–36.

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External Links:

ISSN 0861-1459, MathReview, ZentralBlatt

Referenced by:

§15.15, §15.19(i).

Encodings:

BibTeX

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Generalized functions. Mathematics in Science and Engineering, Vol. 171, Academic Press, Inc., Orlando, FL.

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Notes:

Theory and technique

External Links:

ISBN 0-12-396560-8, MathReview (Qing Yi Chen), ZentralBlatt

Referenced by:

§1.16(viii).

Encodings:

BibTeX

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Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.13(iii), §8.22(ii).

Encodings:

BibTeX

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Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument. Comput. Phys. Comm. 4, pp. 221–226.

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Notes:

Single-precision Fortran, accuracy 12-14S

External Links:

Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

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The Painlevé-Kowalevski and poly-Painlevé tests for integrability. Stud. Appl. Math. 86 (2), pp. 87–165.

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External Links:

ISSN 0022-2526, MathReview (Walter Strampp), ZentralBlatt

Referenced by:

§32.2(i).

Encodings:

BibTeX

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23: 27.18 Methods of Computation: Primes

… ►An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … ►These algorithms are used for testing primality of Mersenne numbers, $2^n-1$, and Fermat numbers, $2^{2^n}+1$. … ►The APR (Adleman–Pomerance–Rumely) algorithm for primality testing is based on Jacobi sums. … ►The AKS (Agrawal–Kayal–Saxena) algorithm is the first deterministic, polynomial-time, primality test. …

24: 1.8 Fourier Series

… ►Formally, if $f(x)$ is a real- or complex-valued $2\pi$-periodic function, … ►Let $f(x)$ be an absolutely integrable function of period $2\pi$, and continuous except at a finite number of points in any bounded interval. … ►For other tests for convergence see Titchmarsh (1962b, pp. 405–410). … ►If $a_n$ and $b_n$ are the Fourier coefficients of a piecewise continuous function $f(x)$ on $[0,2\pi ]$, then … ►If a function $f(x)\in C^2[0,2\pi ]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime }(x)$. …

25: 3.5 Quadrature

… ►which depends on function values computed previously. … ►For effective testing of Gaussian quadrature rules see Gautschi (1983). … ►

Gauss Formula for a Logarithmic Weight Function

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Example

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26: 1.9 Calculus of a Complex Variable

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Differentiation

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Analyticity

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Harmonic Functions

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Weierstrass $M$-test

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