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21: 27.3 Multiplicative Properties
§27.3 Multiplicative Properties
►Except for , , , and , the functions in §27.2 are multiplicative, which means and … ►
27.3.2
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27.3.6
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27.3.10
22: 27.1 Special Notation
§27.1 Special Notation
… ►positive integers (unless otherwise indicated). | |
… | |
prime numbers (or primes): integers () with only two positive integer divisors, and the number itself. | |
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real numbers. | |
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23: 27.12 Asymptotic Formulas: Primes
§27.12 Asymptotic Formulas: Primes
… ►Prime Number Theorem
… ►The number of such primes not exceeding is … ►There are infinitely many Carmichael numbers.24: 27.2 Functions
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§27.2(i) Definitions
… ►where are the distinct prime factors of , each exponent is positive, and is the number of distinct primes dividing . … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ► … ►§27.2(ii) Tables
…25: 24.5 Recurrence Relations
§24.5 Recurrence Relations
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24.5.3
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24.5.5
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§24.5(ii) Other Identities
… ►§24.5(iii) Inversion Formulas
…26: 24.6 Explicit Formulas
27: 27.13 Functions
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§27.13(i) Introduction
►Whereas multiplicative number theory is concerned with functions arising from prime factorization, additive number theory treats functions related to addition of integers. …The subsections that follow describe problems from additive number theory. … ►§27.13(ii) Goldbach Conjecture
… ►§27.13(iii) Waring’s Problem
…28: 26.13 Permutations: Cycle Notation
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►The Stirling cycle numbers of the first kind, denoted by , count the number of permutations of with exactly cycles.
They are related to Stirling numbers of the first kind by
…See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations.
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►The derangement number, , is the number of elements of with no fixed points:
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►A permutation is even or odd according to the parity of the number of transpositions.
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29: 27.9 Quadratic Characters
§27.9 Quadratic Characters
►For an odd prime , the Legendre symbol is defined as follows. … ►
27.9.2
►If are distinct odd primes, then the quadratic reciprocity law states that
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►If an odd integer has prime factorization , then the Jacobi symbol
is defined by , with .
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30: 24.21 Software
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