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1: 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. …Table III lists all solutions n 10 4 of the equation d ( n ) = m , and Table IV lists all solutions n of the equation σ ( n ) = m for all m 10 4 . …6 lists ϕ ( n ) , d ( n ) , and σ ( n ) for n 1000 ; Table 24. …
2: 27.2 Functions
27.2.9 d ( n ) = d | n 1
It is the special case k = 2 of the function d k ( n ) that counts the number of ways of expressing n as the product of k factors, with the order of factors taken into account. …Note that σ 0 ( n ) = d ( n ) . … 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 . …
Table 27.2.2: Functions related to division.
n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n )
3: 27.3 Multiplicative Properties
27.3.1 f ( m n ) = f ( m ) f ( n ) , ( m , n ) = 1 .
27.3.5 d ( n ) = r = 1 ν ( n ) ( 1 + a r ) ,
27.3.8 ϕ ( m ) ϕ ( n ) = ϕ ( m n ) ϕ ( ( m , n ) ) / ( m , n ) .
4: 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 ) ,
5: 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 ) .
6: 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. …
7: 27.14 Unrestricted Partitions
27.14.7 n p ( n ) = k = 1 n σ 1 ( k ) p ( n - k ) ,
27.14.10 A k ( n ) = h = 1 ( h , k ) = 1 k exp ( π i s ( h , k ) - 2 π i n h k ) ,
27.14.19 τ ( m ) τ ( n ) = d | ( m , n ) d 11 τ ( m n d 2 ) , m , n = 1 , 2 , .
27.14.20 τ ( n ) σ 11 ( n ) ( mod 691 ) .
8: 27.10 Periodic Number-Theoretic Functions
Examples are the Dirichlet characters (mod k ) and the greatest common divisor ( n , k ) regarded as a function of n . … 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 .
9: 27.5 Inversion Formulas
§27.5 Inversion Formulas
The multiplicative functions are a subgroup of this group. Generating functions yield many relations connecting number-theoretic functions. …which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: …
10: 27.13 Functions
27.13.6 ( ϑ ( x ) ) 2 = 1 + 4 n = 1 ( δ 1 ( n ) - δ 3 ( n ) ) x n ,