Euler%20sums
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1: 24.20 Tables
§24.20 Tables
►Abramowitz and Stegun (1964, Chapter 23) includes exact values of , , ; , , , , 20D; , , 18D. ►Wagstaff (1978) gives complete prime factorizations of and for and , respectively. …2: 27.2 Functions
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
…It can be expressed as a sum over all primes :
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►This is the number of positive integers that are relatively prime to ; is Euler’s totient.
►If , then the Euler–Fermat theorem states that
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►Table 27.2.2 tabulates the Euler totient function , the divisor function (), and the sum of the divisors (), for .
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3: 24.2 Definitions and Generating Functions
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§24.2(ii) Euler Numbers and Polynomials
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24.2.10
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§24.2(iii) Periodic Bernoulli and Euler Functions
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5: 25.12 Polylogarithms
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►The notation was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828):
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►The cosine series in (25.12.7) has the elementary sum
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►Sometimes the factor is omitted.
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6: 25.11 Hurwitz Zeta Function
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§25.11(iii) Representations by the Euler–Maclaurin Formula
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25.11.10
, .
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§25.11(xi) Sums
… ►For further sums see Prudnikov et al. (1990, pp. 396–397) and Hansen (1975, pp. 358–360). …7: 5.11 Asymptotic Expansions
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►The scaled gamma function is defined in (5.11.3) and its main property is as in the sector .
Wrench (1968) gives exact values of up to .
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►If the sums in the expansions (5.11.1) and (5.11.2) are terminated at () and is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign.
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5.11.13
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5.11.19
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8: 25.6 Integer Arguments
9: 8.17 Incomplete Beta Functions
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►where, as in §5.12, denotes the beta function:
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8.17.3
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§8.17(vi) Sums
►For sums of infinite series whose terms involve the incomplete beta function see Hansen (1975, §62). … ►
8.17.24
positive integers; .
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