sums
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11: 1.7 Inequalities
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§1.7(i) Finite Sums
… ►Cauchy–Schwarz Inequality
… ►Minkowski’s Inequality
… ►Cauchy–Schwarz Inequality
… ►Minkowski’s Inequality
…12: 27.6 Divisor Sums
§27.6 Divisor Sums
►Sums of number-theoretic functions extended over divisors are of special interest. … ►
27.6.1
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►Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors.
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27.6.6
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13: 24.6 Explicit Formulas
14: 16.20 Integrals and Series
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15: 27.1 Special Notation
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positive integers (unless otherwise indicated). | |
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, | sum, product taken over divisors of . |
sum taken over , and relatively prime to . | |
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, | sum, product extended over all primes. |
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16: 27.7 Lambert Series as Generating Functions
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27.7.1
►If , then the quotient is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series:
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27.7.2
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27.7.5
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27.7.6
17: 34.13 Methods of Computation
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►Methods of computation for and symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981).
►For symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989).
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18: 10.44 Sums
§10.44 Sums
►§10.44(i) Multiplication Theorem
… ►§10.44(ii) Addition Theorems
… ►§10.44(iii) Neumann-Type Expansions
… ►§10.44(iv) Compendia
…19: 27.5 Inversion Formulas
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27.5.1
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►which, in turn, is the basis for the Möbius inversion formula relating sums over divisors:
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27.5.3
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27.5.4
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27.5.6
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