multiplicative_fncs

multiplicative_fncs - Multiplicative Functions: NumThy...

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Multiplicative Functions : NumThy Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA squash@ufl.edu Webpage http://www.math.ufl.edu/ squash/ 23 March, 2011 (at 10:41 ) Aside. In number_theory.ams.tex there is a bit more on multiplicative functions also in mult. convolution.ams.tex ; run collect.ams.tex though. See generating_func.latex for an application of the obius fnc. Basic Given two arbitrary 1 functions f,g : Z + C , de- fine their convolution ( “Dirichlet convolution” ) by h f ~ g i ( K ) := X a · b = K f ( a ) · g ( b ) . Each such sum is to be interpreted as over all ordered pairs ( a,b ) of positive divisors of K . Eas- ily, convolution is commutative 1 and associative. Let b G be the set of functions f : Z + C such that f (1) 6 = 0. Inside is G b G , the set of good functions , which have f (1) = 1. Say that a good f is multiplicative if for all posints 2 K and Γ : K Γ = f ( K · Γ ) = f ( K ) · f ( Γ ) . 1 Indeed, they could map into a general ring. If this ring is commutative then convolution will be commutative. 2 Use N to mean “congruent mod N ”. Let n k mean that n and k are co-prime. Use k •| n for “ k divides n ”. Its negation k ± r | n means “ k does not divide n .” Use n |• k and n r | ± k for “ n is/is-not a multiple of k .” Finally, for p a prime and E a natnum: Use double-verticals, p E •|| n , to mean that E is the highest power of p which divides n . Or write n ||• p E to emphasize that this is an assertion about n . Use PoT for Power of Two and PoP for Power of ( a ) Prime . For N a posint, let Φ( N ) mean the set { r [ 1 ..N ] | r N } . The cardinality ϕ ( N ) := | Φ( N ) | is the Euler phi function . (So ϕ ( N ) is the cardinality of the multiplicative group, Φ( N ), in the Z N ring.) Use EFT for the Euler- Fermat Thm , which says: Suppose that integers b N , with N positive. Then b ϕ ( N ) N 1 . Let M G be the set of multiplicative 3 func- tions. Basic functions. Define fncs δ, 1 ,Id M by: δ (1) := 1 and δ ( 6 =1) := 0; 1 ( n ) := 1 and Id ( n ) := n for all posints n . Evidently δ () is a neutral element for convolution. Also define, as n varies over the posints, these MFs: τ ( n ) := X d : d •| n 1 and σ ( n ) := X d : d •| n d. This τ is called the ( number of ) divisor fnc, and σ is called the sum of divisor fnc. So σ (4) = 1 + 2 + 4 = 7 and τ (4) = 1 + 1 + 1 = 3. Each of δ, 1 ,Id, τ , σ is multiplicative. So is Eu- ler ϕ ; this will follow from (4). 1.0 : Theorem. ( b G , ~ ) is a commutative group and M G b G are subgroups. Proof. ( The arguments showing G a group apply to show b G a group. So we only argue for G and M . ) Easily G is sealed under convolution. To show the same for M , take fncs f,h M and posints K Γ . Given posints ( x,y ) with x · y equaling
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multiplicative_fncs - Multiplicative Functions: NumThy...

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