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1: 27.6 Divisor Sums
§27.6 Divisor Sums
Sums of number-theoretic functions extended over divisors are of special interest. … Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. …
2: 27.10 Periodic Number-Theoretic Functions
It can also be expressed in terms of the Möbius function as a divisor sum:
27.10.6 s k ( n ) = d | ( n , k ) f ( d ) g ( k d )
27.10.8 a k ( m ) = d | ( m , k ) g ( d ) f ( k d ) d k .
3: 27.1 Special Notation
d , k , m , n positive integers (unless otherwise indicated).
d | n , d | n sum, product taken over divisors of n .
4: 27.5 Inversion Formulas
which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: …
5: 27.21 Tables
Glaisher (1940) contains four tables: Table I tabulates, for all n 10 4 : (a) the canonical factorization of n into powers of primes; (b) the Euler totient ϕ ( n ) ; (c) the divisor function d ( n ) ; (d) the sum σ ( n ) of these divisors. …
6: 27.3 Multiplicative Properties
27.3.6 σ α ( n ) = r = 1 ν ( n ) p r α ( 1 + a r ) 1 p r α 1 , α 0 .
27.3.7 σ α ( m ) σ α ( n ) = d | ( m , n ) d α σ α ( m n d 2 ) ,
7: 27.2 Functions
27.2.6 ϕ k ( n ) = ( m , n ) = 1 m k ,
27.2.9 d ( n ) = d | n 1
27.2.10 σ α ( n ) = d | n d α ,
is the sum of the α th powers of the divisors of n , where the exponent α can be real or complex. … Table 27.2.2 tabulates the Euler totient function ϕ ( n ) , the divisor function d ( n ) ( = σ 0 ( n ) ), and the sum of the divisors σ ( n ) ( = σ 1 ( n ) ), for n = 1 ( 1 ) 52 . …
8: 27.7 Lambert Series as Generating Functions
27.7.5 n = 1 n α x n 1 x n = n = 1 σ α ( n ) x n ,
9: 27.11 Asymptotic Formulas: Partial Sums
§27.11 Asymptotic Formulas: Partial Sums
27.11.2 n x d ( n ) = x ln x + ( 2 γ 1 ) x + O ( x ) ,
27.11.3 n x d ( n ) n = 1 2 ( ln x ) 2 + 2 γ ln x + O ( 1 ) ,
27.11.4 n x σ 1 ( n ) = π 2 12 x 2 + O ( x ln x ) .
27.11.5 n x σ α ( n ) = ζ ( α + 1 ) α + 1 x α + 1 + O ( x β ) , α > 0 , α 1 , β = max ( 1 , α ) .
10: 27.4 Euler Products and Dirichlet Series
27.4.10 n = 1 d k ( n ) n s = ( ζ ( s ) ) k , s > 1 ,
27.4.11 n = 1 σ α ( n ) n s = ζ ( s ) ζ ( s α ) , s > max ( 1 , 1 + α ) ,