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11: 25.1 Special Notation

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$k,m,n$	nonnegative integers.
$p$	prime number.
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12: 27.11 Asymptotic Formulas: Partial Sums

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27.11.8 $$\sum _{p\le x}\frac{1}{p}=\ln \ln x+A+O\left(\frac{1}{\ln x}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\ln z$: principal branch of logarithm function, $p,p_1,\mathrm{\dots }$: prime numbers, $x$: real number and $A$: constant

Permalink:

http://dlmf.nist.gov/27.11.E8

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 §27.11 and Ch.27

… ►where $\left(h,k\right)=1$, $k>0$. … ►

27.11.15 $$\underset{x\to \mathrm{\infty }}{lim}\sum _{n\le x}\frac{\mu \left(n\right)\ln n}{n}=-1.$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.1 III

Referenced by:

§27.11, §27.11

Permalink:

http://dlmf.nist.gov/27.11.E15

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 §27.11 and Ch.27

►Each of (27.11.13)–(27.11.15) is equivalent to the prime number theorem (27.2.3). The prime number theorem for arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if $\left(h,k\right)=1$, then the number of primes $p\le x$ with $p\equiv h(modk)$ is asymptotic to $x/(\varphi \left(k\right)\ln x)$ as $x\to \mathrm{\infty }$.

13: 6.16 Mathematical Applications

… ►If we assume Riemann’s hypothesis that all nonreal zeros of $\zeta \left(s\right)$ have real part of $\frac{1}{2}$ (§25.10(i)), then ►

6.16.5 $$\mathrm{li}\left(x\right)-\pi (x)=O\left(\sqrt{x}\ln x\right),$$ $x\to \mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $x$: real variable and $\pi (x)$: number of primes $\le x$

A&S Ref:

5.1.50

Permalink:

http://dlmf.nist.gov/6.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(ii), §6.16 and Ch.6

►where $\pi (x)$ is the number of primes less than or equal to $x$. …

14: 25.15 Dirichlet $L$-functions

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25.15.2 $$L(s,\chi )=\prod _p{\left(1-\frac{\chi (p)}{p^s}\right)}^{-1},$$ $\mathrm{\Re }s>1$,

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\mathrm{\Re }$: real part, $p$: prime number, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

infinite product, product representation

Source:

Apostol (1976, p. 231)

Permalink:

http://dlmf.nist.gov/25.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

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25.15.4 $$L(s,\chi )=L(s,{\chi }_0)\prod _{p|k}\left(1-\frac{{\chi }_0(p)}{p^s}\right),$$

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $k$: nonnegative integer, $p$: prime number, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

product representation

Source:

Apostol (1976, p. 262)

Referenced by:

§25.15(i)

Permalink:

http://dlmf.nist.gov/25.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

… ►Related results are: …

15: Tom M. Apostol

… ►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). …

16: 25.16 Mathematical Applications

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§25.16(i) Distribution of Primes

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25.16.1 $$\psi \left(x\right)=\sum _{m=1}^{\mathrm{\infty }}\sum _{p^m\le x}\ln p,$$

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Defines:

$\psi \left(x\right)$: Chebyshev $\psi$-function

Symbols:

$\ln z$: principal branch of logarithm function, $m$: nonnegative integer, $p$: prime number and $x$: real variable

Keywords:

definition, double series, infinite series, series representation

Source:

Apostol (1976, p. 75)

Permalink:

http://dlmf.nist.gov/25.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(i), §25.16 and Ch.25

… ►The prime number theorem (27.2.3) is equivalent to the statement …

17: 24.10 Arithmetic Properties

… ►Here and elsewhere in §24.10 the symbol $p$ denotes a prime number. ►

24.10.1 $$B_{2n}+\sum _{(p-1)|2n}\frac{1}{p}=\text{integer},$$

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Symbols:

$B_n$: Bernoulli numbers, $n$: integer and $p$: prime

Permalink:

http://dlmf.nist.gov/24.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.10(i), §24.10 and Ch.24

… ►The denominator of $B_{2n}$ is the product of all these primes $p$. … ►

24.10.5 $$E_n\equiv E_{n+p-1}(modp),$$

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Symbols:

$E_n$: Euler numbers, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.10(ii)

Permalink:

http://dlmf.nist.gov/24.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.10(ii), §24.10 and Ch.24

… ►Let $B_{2n}=N_{2n}/D_{2n}$, with $N_{2n}$ and $D_{2n}$ relatively prime and $D_{2n}>0$. …

18: 25.10 Zeros

… ►Also, $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re }s=1$, a property first established in Hadamard (1896) and de la Vallée Poussin (1896a, b) in the proof of the prime number theorem (25.16.3). …

19: Bibliography N

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The Development of Prime Number Theory: From Euclid to Hardy and Littlewood. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-66289-8, MathReview (D. R. Heath-Brown), ZentralBlatt

Referenced by:

§27.12.

Encodings:

BibTeX

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20: 27.22 Software

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Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below ${10}^{16}$. Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard $p-1$, and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

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GIMPS. This includes updates of the largest known Mersenne prime.

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Number Theory Web. References and links to software for factorization and primality testing.

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Prime Pages. Information on primes, primality testing, and factorization including links to programs and lists of primes.

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