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1: 24.1 Special Notation
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 ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
2: 27.2 Functions
This is the number of positive integers n that are relatively prime to n ; ϕ ( n ) is Euler’s totient. … and if ϕ ( n ) is the smallest positive integer f such that a f 1 ( mod n ) , then a is a primitive root mod n . The ϕ ( n ) numbers a , a 2 , , a ϕ ( n ) are relatively prime to n and distinct (mod n ). …Note that J 1 ( n ) = ϕ ( n ) . … Table 27.2.2 tabulates the Euler totient function ϕ ( n ) , the divisor function d ( n ) ( = σ 0 ( n ) ), and the sum of the divisors σ ( n ) ( = σ 1 ( n ) ), for n = 1 ( 1 ) 52 . …
3: 27.16 Cryptography
Thus, y x r ( mod n ) and 1 y < n . … To do this, let s denote the reciprocal of r modulo ϕ ( n ) , so that r s = 1 + t ϕ ( n ) for some integer t . (Here ϕ ( n ) is Euler’s totient27.2).) By the Euler–Fermat theorem (27.2.8), x ϕ ( n ) 1 ( mod n ) ; hence x t ϕ ( n ) 1 ( mod n ) . But y s x r s x 1 + t ϕ ( n ) x ( mod n ) , so y s is the same as x modulo n . …
4: 27.3 Multiplicative Properties
27.3.3 ϕ ( n ) = n p | n ( 1 p 1 ) ,
27.3.8 ϕ ( m ) ϕ ( n ) = ϕ ( m n ) ϕ ( ( m , n ) ) / ( m , n ) .
5: 27.6 Divisor Sums
Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. …
27.6.5 d | n | μ ( d ) | ϕ ( d ) = n ϕ ( n ) ,
27.6.6 d | n ϕ k ( d ) ( n d ) k = 1 k + 2 k + + n k ,
6: 27.21 Tables
Glaisher (1940) contains four tables: Table I tabulates, for all n 10 4 : (a) the canonical factorization of n into powers of primes; (b) the Euler totient ϕ ( n ) ; (c) the divisor function d ( n ) ; (d) the sum σ ( n ) of these divisors. Table II lists all solutions n of the equation f ( n ) = m for all m 2500 , where f ( n ) is defined by (27.14.2). …6 lists ϕ ( n ) , d ( n ) , and σ ( n ) for n 1000 ; Table 24. …
7: 27.8 Dirichlet Characters
For any character χ ( mod k ) , χ ( n ) 0 if and only if ( n , k ) = 1 , in which case the Euler–Fermat theorem (27.2.8) implies ( χ ( n ) ) ϕ ( k ) = 1 . There are exactly ϕ ( k ) different characters (mod k ), which can be labeled as χ 1 , , χ ϕ ( k ) . …
27.8.6 r = 1 ϕ ( k ) χ r ( m ) χ ¯ r ( n ) = { ϕ ( k ) , m n ( mod k ) , 0 , otherwise .
8: 27.11 Asymptotic Formulas: Partial Sums
where γ is Euler’s constant (§5.2(ii)). … where γ again is Euler’s constant. …
27.11.6 n x ϕ ( n ) = 3 π 2 x 2 + O ( x ln x ) .
27.11.7 n x ϕ ( n ) n = 6 π 2 x + O ( ln x ) .
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 ( h , k ) = 1 , then the number of primes p x with p h ( mod k ) is asymptotic to x / ( ϕ ( k ) ln x ) as x .
9: 27.5 Inversion Formulas
27.5.4 n = d | n ϕ ( d ) ϕ ( n ) = d | n d μ ( n d ) ,
10: 27.4 Euler Products and Dirichlet Series
§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. If f ( n ) is completely multiplicative, then each factor in the product is a geometric series and the Euler product becomes … Euler products are used to find series that generate many functions of multiplicative number theory. …