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1: 27.3 Multiplicative Properties
Except for ν ( n ) , Λ ( n ) , p n , and π ( x ) , the functions in §27.2 are multiplicative, which means f ( 1 ) = 1 and …
27.3.2 f ( n ) = r = 1 ν ( n ) f ( p r a r ) .
27.3.5 d ( n ) = r = 1 ν ( n ) ( 1 + a r ) ,
27.3.6 σ α ( n ) = r = 1 ν ( n ) p r α ( 1 + a r ) - 1 p r α - 1 , α 0 .
27.3.10 f ( n ) = r = 1 ν ( n ) ( f ( p r ) ) a r .
2: 27.12 Asymptotic Formulas: Primes
§27.12 Asymptotic Formulas: Primes
π ( x ) is the number of primes less than or equal to x . …
Prime Number Theorem
π ( x ) - li ( x ) changes sign infinitely often as x ; see Littlewood (1914), Bays and Hudson (2000). … The largest known prime (2018) is the Mersenne prime 2 82 , 589 , 933 - 1 . …
3: 27.2 Functions
where p 1 , p 2 , , p ν ( n ) are the distinct prime factors of n , each exponent a r is positive, and ν ( n ) is the number of distinct primes dividing n . ( ν ( 1 ) is defined to be 0.) …There is great interest in the function π ( x ) that counts the number of primes not exceeding x . … (See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) …
§27.2(ii) Tables
4: 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 π ( x ) 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 n . …
5: 27.9 Quadratic Characters
§27.9 Quadratic Characters
27.9.2 ( 2 | p ) = ( - 1 ) ( p 2 - 1 ) / 8 .
If p , q are distinct odd primes, then the quadratic reciprocity law states that
27.9.3 ( p | q ) ( q | p ) = ( - 1 ) ( p - 1 ) ( q - 1 ) / 4 .
If an odd integer P has prime factorization P = r = 1 ν ( n ) p r a r , then the Jacobi symbol ( n | P ) is defined by ( n | P ) = r = 1 ν ( n ) ( n | p r ) a r , with ( n | 1 ) = 1 . …
6: 27.21 Tables
§27.21 Tables
Bressoud and Wagon (2000, pp. 103–104) supplies tables and graphs that compare π ( x ) , x / ln x , and li ( x ) . …
7: 27.6 Divisor Sums
27.6.2 d | n μ ( d ) f ( d ) = p | n ( 1 - f ( p ) ) , n > 1 .
27.6.3 d | n | μ ( d ) | = 2 ν ( n ) ,
8: 27.1 Special Notation
d , k , m , n positive integers (unless otherwise indicated).
p , p 1 , p 2 , prime numbers (or primes): integers ( > 1 ) with only two positive integer divisors, 1 and the number itself.
9: 27.4 Euler Products and Dirichlet Series
27.4.1 n = 1 f ( n ) = p ( 1 + r = 1 f ( p r ) ) ,
27.4.2 n = 1 f ( n ) = p ( 1 - f ( p ) ) - 1 .
27.4.3 ζ ( s ) = n = 1 n - s = p ( 1 - p - s ) - 1 , s > 1 .
27.4.9 n = 1 2 ν ( n ) n - s = ( ζ ( s ) ) 2 ζ ( 2 s ) , s > 1 ,
10: 27.16 Cryptography
§27.16 Cryptography
The primes are kept secret but their product n = p q , an 800-digit number, is made public. …