prime numbers in arithmetic progression

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1: 24.1 Special Notation

… ►

Bernoulli Numbers and Polynomials

►The origin of the notation

B_{n}

B_{n}\left(x\right)

, is not clear. … ►

Euler Numbers and Polynomials

… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations

E_{n}

E_{n}\left(x\right)

, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

2: 19.8 Quadratic Transformations

… ►

§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)

►When

a_{0}

and

g_{0}

are positive numbers, define …As

n\to\infty

a_{n}

and

g_{n}

converge to a common limit

M\left(a_{0},g_{0}\right)

called the AGM (Arithmetic-Geometric Mean) of

a_{0}

and

g_{0}

. …showing that the convergence of

c_{n}

to 0 and of

a_{n}

and

g_{n}

M\left(a_{0},g_{0}\right)

is quadratic in each case. … ►where

a_{0}=1

g_{0}=k^{\prime}

p_{0}^{2}=1-\alpha^{2}

Q_{0}=1

, and …

3: 27.11 Asymptotic Formulas: Partial Sums

… ►Dirichlet’s divisor problem (unsolved as of 2022) is to determine the least number

\theta_{0}

such that the error term in (27.11.2) is

O\left(x^{\theta}\right)

for all

\theta>\theta_{0}

. … ►where

\left(h,k\right)=1

k>0

. ►Letting

x\to\infty

in (27.11.9) or in (27.11.11) we see that there are infinitely many primes

p\equiv h\pmod{k}

h, k

are coprime; this is Dirichlet’s theorem on primes in arithmetic progressions. … ►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\leq x

with

p\equiv h\pmod{k}

is asymptotic to

x/(\phi\left(k\right)\ln x)

x\to\infty

4: 27.17 Other Applications

§27.17 Other Applications

►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. ►Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. … ►There are also applications of number theory in many diverse areas, including physics, biology, chemistry, communications, and art. Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.

5: 27.18 Methods of Computation: Primes

§27.18 Methods of Computation: Primes

►An overview of methods for precise counting of the number of primes not exceeding an arbitrary integer

x

is given in Crandall and Pomerance (2005, §3.7). … Oliveira e Silva has calculated

\pi\left(x\right)

for

x=10^{23}

, using the combinatorial methods of Lagarias et al. (1985) and Deléglise and Rivat (1996); see Oliveira e Silva (2006). An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … ►For small values of

n

, primality is proven by showing that

n

is not divisible by any prime not exceeding

\sqrt{n}

. …

6: 27.2 Functions

… ►where

p_{1},p_{2},\dots,p_{\nu\left(n\right)}

are the distinct prime factors of

n

, each exponent

a_{r}

is positive, and

\nu\left(n\right)

is the number of distinct primes dividing

n

. …There is great interest in the function

\pi\left(x\right)

that counts the number of primes not exceeding

x

. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) ►This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the prime number theorem. … ►

§27.2(ii) Tables

…

7: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

… ►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}=\ifrac{N_{2n}}{D_{2n}}

, with

N_{2n}

and

D_{2n}

relatively prime and

D_{2n}>0

. … ►

§24.10(iv) Factors

…

8: 25.15 Dirichlet $L$ -functions

… ►

25.15.2

L\left(s,\chi\right)=\prod_{p}\left(1-\frac{\chi(p)}{p^{s}}\right)^{-1},

\Re s>1

ⓘ

Symbols:: $L\left(\NVar{s},\NVar{\chi}\right)$ : Dirichlet $L$ -function, $\Re$ : 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

►with the product taken over all primes

p

, beginning with

p=2

. … ►There are also infinitely many zeros in the critical strip

0\leq\Re s\leq 1

, located symmetrically about the critical line

\Re s=\frac{1}{2}

, but not necessarily symmetrically about the real axis. …This result plays an important role in the proof of Dirichlet’s theorem on primes in arithmetic progressions (§27.11). Related results are: …

9: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

►The Chinese remainder theorem states that a system of congruences

x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{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. …Their product

m

has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. …

10: 27.12 Asymptotic Formulas: Primes

§27.12 Asymptotic Formulas: Primes

… ►

\pi\left(x\right)

is the number of primes less than or equal to

x