Euler pentagonal number theorem
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11: 27.11 Asymptotic Formulas: Partial Sums
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►where is Euler’s constant (§5.2(ii)).
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►where again is Euler’s constant.
…where , .
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►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 , then the number of primes with is asymptotic to as .
12: 25.16 Mathematical Applications
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►The prime number theorem (27.2.3) is equivalent to the statement
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§25.16(ii) Euler Sums
►Euler sums have the form … ► has a simple pole with residue () at each odd negative integer , . ► is the special case 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 ) 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 … ►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 ): … ►For any character , if and only if , in which case the Euler–Fermat theorem (27.2.8) implies . There are exactly different characters (mod ), which can be labeled as . …If , 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
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24.14.9
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►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
… ► …17: 24.21 Software
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§24.21(ii) , , , and
…18: 24.6 Explicit Formulas
19: 24.5 Recurrence Relations
§24.5 Recurrence Relations
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24.5.4
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24.5.5
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§24.5(ii) Other Identities
… ►§24.5(iii) Inversion Formulas
…20: 24.19 Methods of Computation
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