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completely multiplicative functions

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1: 27.3 Multiplicative Properties
27.3.8 ϕ ( m ) ϕ ( n ) = ϕ ( m n ) ϕ ( ( m , n ) ) / ( m , n ) .
A function f is completely multiplicative if f ( 1 ) = 1 and …
2: 27.20 Methods of Computation: Other Number-Theoretic Functions
For a completely multiplicative function we use the values at the primes together with (27.3.10). …
3: 27.4 Euler Products and Dirichlet Series
The completely multiplicative function f ( n ) = n s gives the Euler product representation of the Riemann zeta function ζ ( s ) 25.2(i)): …
4: 27.8 Dirichlet Characters
If k ( > 1 ) is a given integer, then a function χ ( n ) is called a Dirichlet character (mod k ) if it is completely multiplicative, periodic with period k , and vanishes when ( n , k ) > 1 . …
5: 22.5 Special Values
Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its z -derivative (or at a pole, the residue), for values of z that are integer multiples of K , i K . …
6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Such orthonormal sets are called complete. … , of unit multiplicity, unless otherwise stated. …and completeness implies … and completeness relation …
7: 22.8 Addition Theorems
§22.8 Addition Theorems
§22.8(iii) Special Relations Between Arguments
22.8.22 z 1 + z 2 + z 3 + z 4 = 2 K ( k ) .
If sums/differences of the z j ’s are rational multiples of K ( k ) , then further relations follow. …
8: 23.1 Special Notation
(For other notation see Notation for the Special Functions.)
𝕃 lattice in .
K ( k ) , K ( k ) complete elliptic integrals (§19.2(i)).
n set of all integer multiples of n .
The main functions treated in this chapter are the Weierstrass -function ( z ) = ( z | 𝕃 ) = ( z ; g 2 , g 3 ) ; the Weierstrass zeta function ζ ( z ) = ζ ( z | 𝕃 ) = ζ ( z ; g 2 , g 3 ) ; the Weierstrass sigma function σ ( z ) = σ ( z | 𝕃 ) = σ ( z ; g 2 , g 3 ) ; the elliptic modular function λ ( τ ) ; Klein’s complete invariant J ( τ ) ; Dedekind’s eta function η ( τ ) . …
9: 27.2 Functions
§27.2 Functions
§27.2(i) Definitions
They tend to thin out among the large integers, but this thinning out is not completely regular. … This is Jordan’s function. … This is Liouville’s function. …
10: Bibliography K
  • M. K. Kerimov and S. L. Skorokhodov (1985c) Calculation of the multiple zeros of the derivatives of the cylindrical Bessel functions J ν ( z ) and Y ν ( z ) . Zh. Vychisl. Mat. i Mat. Fiz. 25 (12), pp. 1749–1760, 1918.
  • M. K. Kerimov and S. L. Skorokhodov (1986) On multiple zeros of derivatives of Bessel’s cylindrical functions. Dokl. Akad. Nauk SSSR 288 (2), pp. 285–288 (Russian).
  • M. K. Kerimov and S. L. Skorokhodov (1987) On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions. Zh. Vychisl. Mat. i Mat. Fiz. 27 (11), pp. 1628–1639, 1758.
  • M. K. Kerimov and S. L. Skorokhodov (1988) Multiple complex zeros of derivatives of the cylindrical Bessel functions. Dokl. Akad. Nauk SSSR 299 (3), pp. 614–618 (Russian).
  • S. Koumandos and M. Lamprecht (2010) Some completely monotonic functions of positive order. Math. Comp. 79 (271), pp. 1697–1707.