quintuple product identity
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1: 17.8 Special Cases of Functions
2: 24.10 Arithmetic Properties
…
►The denominator of is the product of all these primes .
…
►where .
…valid when and , where is a fixed integer.
…
►valid for fixed integers , and for all such that
and .
►
24.10.9
…
3: 27.16 Cryptography
…
►The primes are kept secret but their product
, an 800-digit number, is made public.
…
►Thus, and .
…
►By the Euler–Fermat theorem (27.2.8), ; hence .
But , so is the same as modulo .
…
4: 27.15 Chinese Remainder Theorem
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►The Chinese remainder theorem states that a system of congruences , always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod ), where is the product of the moduli.
…
►Their product
has 20 digits, twice the number of digits in the data.
…
5: 15.17 Mathematical Applications
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►
§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 satisfies the identity …In this case the infinite product on the right (extended over all primes ) is also absolutely convergent and is called the Euler product of the series. If is completely multiplicative, then each factor in the product is a geometric series and the Euler product becomes …7: 24.19 Methods of Computation
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►Another method is based on the identities
►
…
►If denotes the right-hand side of (24.19.1) but with the second product taken only for , then for .
…
►For number-theoretic applications it is important to compute for ; in particular to find the irregular pairs
for which .
…
24.19.1
►
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 generates , and generates , then the product
generates
…called the Dirichlet product (or convolution) of and .
The set of all number-theoretic functions with forms an abelian group under Dirichlet multiplication, with the function in (27.2.5) as identity element; see Apostol (1976, p. 129).
…For example, the equation is equivalent to the identity
…
►
27.5.8
…
10: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►A complex linear vector space
is called an inner product space if
an inner product
is defined
for all
with the properties:
(i)
is complex linear in
;
(ii)
;
(iii)
;
(iv) if
then
.
With norm defined by
…
►The inner product of
and
is
…
►Functions
for which
are
identified with each other. The space
becomes a separable
Hilbert space with inner product
…thus generalizing the inner product of (1.18.9).
When
is absolutely continuous, i.e.
, see §1.4(v), where the nonnegative weight function
is
Lebesgue measurable on
. In this section we will only consider the special case
, so
;
in which case
.
…