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Euler pentagonal number theorem

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11: 27.11 Asymptotic Formulas: Partial Sums
where γ is Euler’s constant (§5.2(ii)). … where γ again is Euler’s constant. …where ( h , k ) = 1 , k > 0 . … 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 ( 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 .
12: 25.16 Mathematical Applications
The prime number theorem (27.2.3) is equivalent to the statement …
§25.16(ii) Euler Sums
Euler sums have the form … H ( s ) has a simple pole with residue ζ ( 1 2 r ) ( = B 2 r / ( 2 r ) ) at each odd negative integer s = 1 2 r , r = 1 , 2 , 3 , . H ( s ) is the special case H ( s , 1 ) of the function …
13: 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. …
14: 27.8 Dirichlet Characters
§27.8 Dirichlet Characters
An example is the principal character (mod k ): … 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 ) . …If ( n , k ) = 1 , then the characters satisfy the orthogonality relation
15: 24.14 Sums
§24.14 Sums
§24.14(i) Quadratic Recurrence Relations
§24.14(ii) Higher-Order Recurrence Relations
24.14.9 ( 2 n ) ! ( 2 j ) ! ( 2 k ) ! ( 2 ) ! E 2 j E 2 k E 2 = 1 2 ( E 2 n E 2 n + 2 ) .
For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).
16: 24.2 Definitions and Generating Functions
§24.2 Definitions and Generating Functions
§24.2(i) Bernoulli Numbers and Polynomials
§24.2(ii) Euler Numbers and Polynomials
§24.2(iii) Periodic Bernoulli and Euler Functions
Table 24.2.1: Bernoulli and Euler numbers.
n B n E n
17: 24.21 Software
§24.21(ii) B n , B n ( x ) , E n , and E n ( x )
18: 24.6 Explicit Formulas
§24.6 Explicit Formulas
24.6.4 E 2 n = k = 1 n 1 2 k 1 j = 1 k ( 1 ) j ( 2 k k j ) j 2 n ,
24.6.5 E 2 n = 1 2 n 1 k = 0 n 1 ( 1 ) n k ( n k ) 2 n j = 0 k ( 2 n 2 j k j ) 2 j ,
24.6.10 E n = 1 2 n k = 1 n + 1 ( n + 1 k ) j = 0 k 1 ( 1 ) j ( 2 j + 1 ) n .
24.6.12 E 2 n = k = 0 2 n 1 2 k j = 0 k ( 1 ) j ( k j ) ( 1 + 2 j ) 2 n .
19: 24.5 Recurrence Relations
§24.5 Recurrence Relations
24.5.4 k = 0 n ( 2 n 2 k ) E 2 k = 0 , n = 1 , 2 , ,
24.5.5 k = 0 n ( n k ) 2 k E n k + E n = 2 .
§24.5(ii) Other Identities
§24.5(iii) Inversion Formulas
20: 24.19 Methods of Computation
§24.19(i) Bernoulli and Euler Numbers and Polynomials
Equations (24.5.3) and (24.5.4) enable B n and E n to be computed by recurrence. …A similar method can be used for the Euler numbers based on (4.19.5). … For algorithms for computing B n , E n , B n ( x ) , and E n ( x ) see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180).
§24.19(ii) Values of B n Modulo p