On density and multiplicative structure of sets of generalized - PDF document

On density and multiplicative structure of sets of generalized integers ˇ Stefan Porubsk´ y Institute of Computer Science Academy of Sciences of the Czech Republic Pod Vod´ arenskou vˇ eˇ z´ ı 2 182 07 Prague 8 email: Stefan.Porubsky@cs.cas.cz y, ˇ Porubsk´ S. : Notes on density and multiplicative structure of sets of gener- alized integers , in: Topics in Classical Number Theory , Colloquia Mathematica Societatis J´ anos Bolyai, 34. , Budapest 1984, pp. 1295 – 1315

N A ( x ) = � A ⊂ N , 1 , x > 1 a i ∈ A a i ≤ x • asymptotic density : N A ( x ) N A ( x ) d ( A ) = lim inf , d ( A ) = lim sup x x x →∞ x →∞ • logarithmic density : � � 1 1 1 1 ℓ ( A ) = lim inf a i , ℓ ( A ) = lim sup log x log x a i x →∞ x →∞ a i ∈ A a i ∈ A a i ≤ x a i ≤ x • Schnirelmann density : � � N A ( x ) h ( A ) = inf x x 0 ≤ d ( A ) ≤ ℓ ( A ) ≤ ℓ ( A ) ≤ d ( A ) Common features: a. density is a non–negative real number b. finite sets have zero density c. if A ⊂ B ⊂ N then density of A does not exceed density of B 2

Arithmetical semigroups • ( G , . ) free commutative semigroup with identity ele- ment 1 G • P G ≤ ∞ the set of generators (the so–called primes ) • norm | · | : G → R : ⋆ | 1 G | = 1 , | a | > 1 for all a ∈ G , ⋆ | ab | = | a | . | b | for all a, n ∈ G , ⋆ N G ( x ) = � 1 < ∞ for each real x . | a |≤ x a ∈ G Axiom A: There exists positive constants A and δ and a constant η with 0 ≤ η < δ , such that N G ( x ) = Ax δ + O ( x η ) ζ –function of G : ζ G ( s ) = � 1 | a | s a ∈ G Lemma. Let G be an arithmetical semigroup satis- fying Axiom A. Then � a ∈ G | a | − z is absolutely con- vergent for ℜ ( z ) > δ , and divergent for ℜ ( z ) ≤ δ . Moreover � | a | − δ = δA log x + γ G + O ( x η − δ ) . | a |≤ x 3

Example 1. G = N , the set of positive integers where N N ( x ) = x + O (1) . Example 2. G = G K , the semigroup of all non–zero integral ideals in a given algebraic number field K of degree n = [ K : Q ] over rationals Q with the usual norm function | a | = card ( O K / a ) . Then � � n − 1 N K ( x ) = A K x + O x , n +1 where A K can be explicitly given. Example 3. G = A the category of all finite Abelian groups with the usual direct product operation and the norm | H | = card ( H ) . Fundamental Theorem on finite Abelian groups shows that A is free and that the gen- erators are the cyclic groups of prime–power order. The fact that this arithmetical semigroup satisfies Axiom A follows from an older result of Erd˝ os and Szekeres that N A ( x ) = αx + O ( √ x ) , ∞ � where α = ζ N ( js ) with ζ N = ζ denoting the classi- j =1 cal Riemann zeta function. 4

δ ∈ ( −∞ , + ∞ ) δ –regularly varying function function F ( x ) defined and measurable on � 0 , ∞ ) : F ( λx ) F ( x ) = λ δ lim x →∞ if for every λ > 0 If F ( x ) = x δ L ( x ) , then L ( x ) is slowly oscillating (i.e. δ = 0 ). Arithmetical semigroup G will be called δ –regular if the counting function N G ( x ) is δ –regularly varying function. Lemma. Let G be a δ –regular semigroup. Then � a ∈ G | a | − z is convergent for all ℜ ( z ) > δ and di- vergent for all ℜ ( z ) < δ . x Wegmann (1966) : If N G ( x ) ∼ log 2 x (i.e. G is 1 – regular), then � a ∈ G | a | − 1 < ∞ . 5

m : G → R , C ⊂ G : χ C indicator of C N C ( m , x ) = � | a |≤ x m ( a ) χ C ( a ) C ( m , x ) = � N ′ | a | = x m ( a ) χ C ( a ) � | a |≤ x m ( a ) χ C ( a ) = N C ( m , x ) σ x ( C , m ) = � | a |≤ x m ( a ) N G ( m , x ) lower m –density : σ ( C , m ) = lim inf x →∞ σ x ( C , m ) upper m –density : σ ( C , m ) = lim sup x →∞ σ x ( C , m ) A. m is non–negative, i.e. m ( a ) ≥ 0 for every a ∈ G B. � m ( a ) diverges a ∈ G A implies a B implies b B implies the summation method is regular 6

Knopp’s theorem on convergence kernel: Theorem. Let m and s be two positive functions de- fined on an arithmetical semigroup G such that the series � (i) s ( a ) diverges a ∈ G � m ( a ) N ′ G ( m , x ) | a | = x (ii) lim N G ( m , x ) = lim � m ( a ) = 0 x →∞ x →∞ | a |≤ x � s ( a ) N ′ G ( s , x ) | a | = x lim N G ( s , x ) = lim � s ( a ) = 0 x →∞ x →∞ | a |≤ x (iii) if a 1 , a 2 ∈ G be such that | a 1 | ≤ | a 2 | then m ( a 2 ) m ( a 1 ) ≥ s ( a 2 ) s ( a 1 ) . Then σ ( C, m ) ≤ σ ( C, s ) ≤ σ ( C, s ) ≤ σ ( C, m ) for every C ⊂ G . 7

If N G ( m , x ) is δ –regular, (ii) is superfluous: � | a |≤ x m ( n ) = x δ L ( x ) , and 0 < α < 1 arbitrary � � | a | = x m ( n ) αx< | a |≤ x m ( n ) 0 ≤ � | a |≤ x m ( n ) ≤ � | a |≤ x m ( n ) = x δ L ( x ) − α δ x δ L ( αx ) → 1 − α δ x δ L ( x ) M. to every a ∈ G there exists a positive real number m ( a ) , � m < 1 such that for every subset C ⊂ G � having the m –density σ ( C , m ) the set a C = { ac : c ∈ G } has also the m –density and σ ( a C , m ) = � m ( a ) σ ( C , m ) . m ( a ) = 1 • if m ( a ) = 1 for every a ∈ G then � a • if m is completely multiplicative and N G ( m , x ) = x ∆ L ( x ) , where L ( x ) is slowly oscillating, then m ( a ) = m ( a ) | a | − ∆ . � 8

Question: Under which conditions does � m fulfil the conditions A and B ? Theorem. Let G be an arithmetical semigroup. Let m satisfy conditions A , B , and M . Let in the case, when � m ( p ) < ∞ , ( ∗ ) � p ∈ P G we have uniformly in x and p ∈ P G σ p G ( m , x ) = O ( � m ( p )) ( ∗∗ ) Then � m fulfils conditions A and B . Note: ( ∗∗ ) cannot be omitted if ( ∗ ) holds! Take Wegmann’s G with asymptotic density. Then ( ∗ ) m ( p ) = | p | − 1 ), while it can be shown that the holds ( � finite set G � P G � = { a ∈ G : p ∤ a for every p ∈ P G } = { 1 G } has non–zero density � p ∈ P G (1 − | p | − 1 ) . 9

Lemma. If the arithmetical semigroup G satisfies Axiom A then the series � | p | − δ p ∈ P G diverges. We have • � m is completely multiplicative ( ab G = a ( b G ) ), • lim m ( a ) = 0 (if lim | p |→∞ � m ( p ) = 0 ). | a |→∞ � Therefore 1 • a �→ m ( a ) is a norm on G � � � − z ζ G ( z ) = � 1 = � • � m ( a ) z � m ( a ) � a ∈ G a ∈ G B says that � ζ G ( z ) has a pole at z = 1 if m ( a ) = 1 for every a ∈ G and G is δ –regular, then � ζ G ( z ) = ζ G ( sδ ) , and therefore B holds if G satisfies Axiom A. 10

slowly oscillating function L ( x ) is called good if � x 1 L ( y ) y − 1 d y . x →∞ Z ( x ) = ∞ , where Z ( x ) = lim if lim inf x →∞ L ( x ) > 0 , then L is good Theorem. Let m be a completely multiplicative func- tion defined on an arithmetical semigroup G . Let N G ( m , x ) = � m ( a ) = x δ L ( x ) , | a |≤ x with L ( x ) a good slowly oscillating function. Then 0 ≤ σ ( C , m ) ≤ σ ( C , � m ) ≤ σ ( C , � m ) ≤ σ ( C , m ) ≤ 1 for every C ⊂ G . Corollary. Let m be a positive completely multiplica- tive function defined on the arithmetical semigroup G such that � m ( a ) = L ( x ) , | a |≤ x where L is a good slowly oscillating function, then σ ( C , m ) = σ ( C , � m ) , and σ ( C , � m ) = σ ( C , m ) for every C ⊂ G . 11

Corollary. Let m be a completely multiplicative func- tion defined on an arithmetical semigroup G . Let N G ( m , x ) = � m ( a ) = x δ L ( x ) , | a |≤ x with L ( x ) a good slowly oscillating function. Then � m = � m . � Corollary. Let the arithmetical semigroup G satisfies Axiom A. Let m be a positive function defined on G such that • the series � m ( a ) diverges, a ∈ G � m ( a ) N ′ G ( m , x ) | a | = x • lim N G ( m , x ) = lim � m ( a ) = 0 , x →∞ x →∞ | a |≤ x • | a 1 | ≤ | a 2 | ⇒ m ( a 1 ) ≤ m ( a 2 ) then the lower and upper m –density coincides with the lower and upper logarithmic density. 12

Theorem. Let m : G → R + be a completely multi- plicative function such that N G ( m , x ) = Bx ∆ + O ( x Θ ) , 0 ≤ Θ < ∆ as x → ∞ , then m , x ) = ∆ B log x + ψ m + O ( x Θ − ∆ ) N G ( � with a suitable ψ m . lower m –density : � m ( a ) N C ( m , x ) a ∈C , | a |≤ x σ G ( C , m ) = lim inf N G ( m , x ) = lim inf Bx ∆ x →∞ x →∞ upper m –density : � m ( a ) N C ( m , x ) a ∈C , | a |≤ x σ G ( C , m ) = lim sup N G ( m , x ) = lim sup Bx ∆ x →∞ x →∞ lower � m –density : � m ( a ) | a | − ∆ N C ( � m , x ) a ∈C , | a |≤ x d G ( C ) = lim inf m , x ) = lim inf N G ( � ∆ B log x x →∞ x →∞ upper � m –density : � m ( a ) | a | − ∆ N C ( � m , x ) a ∈C , | a |≤ x d G ( C ) = lim sup m , x ) = lim sup N G ( � ∆ B log x x →∞ x →∞

if Axiom A holds for G then � | a | − δ ∼ δA log x , and | a |≤ x N G ( x ) ∼ Ax δ lower logarithmic density : � 1 | a | − δ ℓ G ( C ) = lim inf δA log x x →∞ | a |≤ x upper logarithmic density : 1 � | a | − δ ℓ G ( C ) = lim sup δA log x x →∞ | a |≤ x lower asymptotic density : � 1 N C ( 1 , x ) a ∈C , | a |≤ x d G ( C ) = lim inf N G ( x ) = lim inf Ax δ x →∞ x →∞ upper asymptotic density : � 1 N C ( 1 , x ) a ∈C , | a |≤ x d G ( C ) = lim sup N G ( x ) = lim sup Ax δ x →∞ x →∞ 14