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1: 17.8 Special Cases of ψ r r Functions
Jacobi’s Triple Product
Quintuple Product Identity
2: 24.10 Arithmetic Properties
The denominator of B 2 n is the product of all these primes p . … where m n 0 ( mod p 1 ) . …valid when m n ( mod ( p 1 ) p ) and n 0 ( mod p 1 ) , where ( 0 ) is a fixed integer. … valid for fixed integers ( 1 ) , and for all n ( 1 ) such that 2 n 0 ( mod p 1 ) and p | 2 n .
24.10.9 E 2 n { 0 ( mod p ) if  p 1 ( mod 4 ) , 2 ( mod p ) if  p 3 ( mod 4 ) ,
3: 27.16 Cryptography
The primes are kept secret but their product n = p q , an 800-digit number, is made public. … Thus, y x r ( mod n ) and 1 y < n . … 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: 27.15 Chinese Remainder Theorem
The Chinese remainder theorem states that a system of congruences x a 1 ( mod m 1 ) , , x a k ( mod 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. … Their product m has 20 digits, twice the number of digits in the data. …
5: 15.17 Mathematical Applications
§15.17(iv) Combinatorics
In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …
6: 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. Every multiplicative f satisfies the identity …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 …
7: 24.19 Methods of Computation
Another method is based on the identities
24.19.1 N 2 n = 2 ( 2 n ) ! ( 2 π ) 2 n ( p 1 | 2 n p ) ( p p 2 n p 2 n 1 ) ,
D 2 n = p 1 | 2 n p ,
If N ~ 2 n denotes the right-hand side of (24.19.1) but with the second product taken only for p ( π e ) 1 2 n + 1 , then N 2 n = N ~ 2 n for n 2 . … For number-theoretic applications it is important to compute B 2 n ( mod p ) for 2 n p 3 ; in particular to find the irregular pairs ( 2 n , p ) for which B 2 n 0 ( mod p ) . …
8: 21.6 Products
§21.6 Products
§21.6(i) Riemann Identity
Then …This is the Riemann identity. …Many identities involving products of theta functions can be established using these formulas. …
9: 27.5 Inversion Formulas
If a Dirichlet series F ( s ) generates f ( n ) , and G ( s ) generates g ( n ) , then the product F ( s ) G ( s ) generates …called the Dirichlet product (or convolution) of f and g . The set of all number-theoretic functions f with f ( 1 ) 0 forms an abelian group under Dirichlet multiplication, with the function 1 / n in (27.2.5) as identity element; see Apostol (1976, p. 129). …For example, the equation ζ ( s ) ( 1 / ζ ( s ) ) = 1 is equivalent to the identity
27.5.8 g ( n ) = d | n f ( d ) f ( n ) = d | n ( g ( n d ) ) μ ( d ) .
10: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
A complex linear vector space V is called an inner product space if an inner product u , v is defined for all u , v V with the properties: (i) u , v is complex linear in u ; (ii) u , v = v , u ¯ ; (iii) v , v 0 ; (iv) if v , v = 0 then v = 0 . With norm defined byThe inner product of v and w = ( d 0 , d 1 , d 2 , ) isFunctions f , g L 2 ( X , d α ) for which f g , f g = 0 are identified with each other. The space L 2 ( X , d α ) becomes a separable Hilbert space with inner productthus generalizing the inner product of (1.18.9). When α is absolutely continuous, i.e. d α ( x ) = w ( x ) d x , see §1.4(v), where the nonnegative weight function w ( x ) is Lebesgue measurable on X . In this section we will only consider the special case w ( x ) = 1 , so d α ( x ) = d x ; in which case L 2 ( X ) L 2 ( X , d x ) .