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1: 27.20 Methods of Computation: Other Number-Theoretic Functions
§27.20 Methods of Computation: Other Number-Theoretic Functions
2: 27.17 Other Applications
§27.17 Other Applications
Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. …
3: 6.16 Mathematical Applications
§6.16(ii) Number-Theoretic Significance of li ( x )
4: 27.5 Inversion Formulas
§27.5 Inversion Formulas
The set of all number-theoretic functions f with f ( 1 ) 0 forms an abelian group under Dirichlet multiplication, with the function 1 / n in (27.2.5) as identity element; see Apostol (1976, p. 129). …Generating functions yield many relations connecting number-theoretic functions. …
27.5.2 d | n μ ( d ) = 1 n ,
5: 27.8 Dirichlet Characters
§27.8 Dirichlet Characters
27.8.4 χ ( n ) = 0 , ( n , k ) > 1 .
If ( n , k ) = 1 , then the characters satisfy the orthogonality relation
27.8.6 r = 1 ϕ ( k ) χ r ( m ) χ ¯ r ( n ) = { ϕ ( k ) , m n ( mod k ) , 0 , otherwise .
A divisor d of k is called an induced modulus for χ if …
6: 27.10 Periodic Number-Theoretic Functions
§27.10 Periodic Number-Theoretic Functions
If k is a fixed positive integer, then a number-theoretic function f is periodic (mod k ) if …
7: 27.21 Tables
§27.21 Tables
8: 27.3 Multiplicative Properties
§27.3 Multiplicative Properties
27.3.8 ϕ ( m ) ϕ ( n ) = ϕ ( m n ) ϕ ( ( m , n ) ) / ( m , n ) .
9: 27.6 Divisor Sums
§27.6 Divisor Sums
Sums of number-theoretic functions extended over divisors are of special interest. …
10: 27.7 Lambert Series as Generating Functions
§27.7 Lambert Series as Generating Functions