BETA DISTRIBUTION ON ARITHMETICAL SEMIGROUPS

Transcription

1 Annales Univ Sci Budapest Sect Comp 47 (2018) BETA DISTRIBUTION ON ARITHMETICAL SEMIGROUPS Gintautas Bareikis and Algirdas Mačiulis (Vilnius Lithuania) Communicated by Imre Kátai (Received February ; accepted April ) Abstract The sequences of distributions defined on an arithmetical semigroup are considered We prove that any Beta distribution can occur as a limit law for such sequences 1 Introduction and result Let a b be positive constants The Beta law B(a b) is two parameters distribution concentrated on the interval u [0; 1] with distribution function B(u; a b) := Γ(a + b) u dt Γ(a)Γ(b) t 1 a (1 t) 1 b 0 For the distributions defined via arithmetical functions the convergence to the Beta law was considered in [6] [4] [2] [5] [3] Authors of this paper have proved [1] that the one parameter Beta law B(a 1 a) can be simulated by means of a sequence of the distributions defined on a multiplicative semigroup In this paper we generalize this result for any two parameters Beta law B(a b) Key words and phrases: Arithmetical semigroup multiplicative function divisor function 2010 Mathematics Subject Classification: 11N45 11N80 11K65

2 148 G Bareikis and A Mačiulis To discuss this problem we need some notations Let G be a commutative multiplicative semigroup with identity element a 0 and generated by a countable subset P of prime elements We assume that k l m n are non-negative integers a b d G p P and a completely additive degree function : G N {0} is defined so that (p) 1 for each prime p In the sequel we assume that the semigroup G satisfies ( see [10] [9]) the following Axiom A There exist constants A > 0 q > 1 and 0 ν < 1 such that G(n) := #{a G : (a) = n} = Aq n + O(q νn ) The prime number theorem in semingroup G (see [7] [8]) yields π(n) := #{p P : (p) = n} = qn n (1 ( 1)n I(G)) + O(q µn ) with some max(1/2 ν) < µ < 1 Here I(G) = 1 if Z( 1) = 0 and I(G) = 0 otherwise Here Z(z) := ( ) n z G(n) z < 1 q n 0 is the generating function which has an analytic continuation into the disc z < q 1 ν and Z(z) 0 for z 1 with the possible exception at the point z = 1 Definition 11 Let g : G (0 ) be a multiplicative function such that g(p m ) C for m N any p P and some C > 0 We say that g belongs to the class M(κ C c) κ 0 if the function defined by H(z) := m 1 ( ) m z (g(p) κ) z < 1 q (p)=m has an analytic continuation into the disc z < 1 + c for some c > 0 For a multiplicative function f : G [0 ) and v 0 set T f (a v) := d a (d) v f(d) T f (a) := T f (a (a)) Here and in the following the starred sum or product symbols mean that these operations are used over corresponding elements of the semigroup G Definition 12 We say that a pair (g; f) of the multiplicative functions belongs to the class M(κ α C 1 c 1 ) if g M(κ C 1 c 1 ) and g T f M(α C 1 c 1 )

3 Beta distribution on arithmetical semigroups 149 For any a G and t [0 1] set When G(n) > 0 we define X(a t) := T f (a (a)t) T f (a) (11) F n (t; g f) := q 1 qg n (g) (a) n g(a)x(a t) where g : G (0 ) is a multiplicative function and G n (g) := g(a) (a)=n Note that axiom A implies G(n) > 0 for all sufficiently large n We consider a multiplicative function f(a) defined on the semigroup with axiom A provided the associated divisors function T f (a) satisfies some analytic conditions The aim of our paper is to show that sequence (11) can be approximated by the Beta distribution with some positive parmeters The main result is the following Theorem Suppose that (g f) M(κ α; C 1 c 1 ) If 0 < α < κ then uniformly for 0 t 1 ( 1 F n (t; g f) = B(t; κ α α) + O n κ α + 1 n α + (ln n)ɛ(κ α) + (ln n) ɛ(α) ) n as n Here ɛ(1) = 1 and ɛ(v) = 0 for v 1 Unless otherwise indicated we assume that the implicit constants in the or O() symbols depend at most on the parameters and constants involved in the definitions of the semigroup G and corresponding classes M() or M() 2 Preliminaries We will need the estimate for the sum of the shifted multiplicative functions defined on G M n (g d) := 1 Aq n g(ad) (a)=n The following lemma yields the result of this type

4 150 G Bareikis and A Mačiulis Lemma 21 ([1]) Let g : G [0 ) be a multiplicative function such that g M(κ C c) with some positive constants κ C and c Then uniformly for all d G and n 0 M n (g d) = (A(n + 1)) κ 1 ( L(κ g) g(d) Γ(κ) p ( )) ĝ(d) + O n + 1 where L(κ g) and the multiplicative functions g and ĝ are defined by L(κ g) := ( 1 1 ) κ g(p k ) q (p) q k (p) g(p m ) := ĝ(p m ) := g(p k ) q k (p) k 0 ( 1 + c ) 1 q 2 (p)/3 1 k 0 k 0 k 0 Here c 1 0 is a constant depending on κ and C Lemma 22 ([10] p86) Suppose that σ R Then m=1 g(p k+m ) q k (p) g(p k+m ) q 2k (p)/3 n m σ q m = q q 1 nσ q n + O ( n σ 1 q n) Lemma 23 For 0 t 1 n 1 and γ δ R we have (21) where Moreover 1 (1 + k) γ (1 + n k) δ = T n (t; γ δ) := k nt ( 1 = n 1 γ δ J(t; γ δ n 1 ) + O n δ + 1 n γ J(t; γ δ η) := t 0 dv (η + v) γ (η + 1 v) δ ) (22) T n (t; γ δ) 1 n δ + 1 n γ + (ln n)ɛ(δ) + (ln n) ɛ(γ) n δ+γ 1 In case δ < 1 and γ < 1 we have (23) J(t; γ δ n 1 ) = J(t; γ δ 0) + O ( n γ 1 + n δ 1 + n 1) The implicit constants in the or O() symbols depend on γ and δ only

5 Beta distribution on arithmetical semigroups 151 Proof Easy to check that (24) 1 0 dv (n 1 + v) γ (n v) δ γδ n δ 1 + n γ 1 + ln ɛ(γ) n + ln ɛ(δ) n The first assertion in this lemma follows from Euler-Maclaurin summation formula and (24) The inequality (22) follows from (21) and (24) Formula (23) follows from Lagrange mean value theorem applying the estimate (24) (see in [3] and [1]) 3 Proof of Theorem Assumptions of the Theorem imply that the multiplicative functions h := g/t f M(α C 1 c 1 ) and f h M(κ α C 1 c 1 ) We have (31) F n (t; g f) = S n (t) + R n (t) t [0; 1] where S n (t) := q 1 q G n (g) R n (0) = 0 and for t (0 1] 0 m n (a)=m g(a)t f (a nt) T f (a) R n (t) := q 1 q G n (g) 1 G n (g) (a) n g(a) f(d) (a)=m T f (a nt) T f (n (a)t) T f (a) k (d)(1 t)/t (a)=k Applying Lemma 21 for the inner sum we have (32) (a)=k h(ad) = A α q k (1 + k) 1 α ( L(α h) h(d) + O Γ(α) h(ad) An easy calculation shows that ( ) h(p m ) = h(p m 1 ) + O T f (p m )q (p) ( ) ĥ(p m ) = h(p m 1 ) + O T f (p m )q 2 (p)/3 ( )) ĥ(d) 1 + k

6 152 G Bareikis and A Mačiulis Note that these relations imply f h fĥ M(κ α C 1 c 1 ) Using (32) we get R n (t) 1 G n (g) (d) tn f(d)ĥ(d) k (d)(1 t)/t q k (1 + k) 1 α By Lemma 22 (33) R n (t) q n q nt G n (g)(1 + n(1 t)) 1 α m nt (a)=m f(a)ĥ(a) Since g M(κ C 1 c 1 ) Lemma 21 yields (34) G n (g) = A α q n (1 + n) 1 κ ( L(κ h) Γ(κ) ( )) 1 + O 1 + n For the inner sum in (33) we can apply Lemma 21 with d = a 0 Then employing Lemma 22 again and having in mind (34) we obtain Thus (31) becomes R n (t) (1 + n) 1 κ ( (1 + n(1 t)) α 1 (1 + nt) κ α 1) (35) F n (t g f) = S n (t) + O ( n α κ + n α + n 1) uniformly for t [0 1] It remains to evaluate the sum S n (t) Changing order of summation we obtain S n (t) = q 1 n (d) f(d) h(ad) q G n (g) Hence (32) gives m=0 (a)=m (36) S n (t) = Φ n (t) + O (r(n t)) where Φ n (t) := (q 1)Aα L(α h) q G n (g)γ(α) n (d) q m f(d) h(d) (1 + m) 1 α m=0 and r(n t) := 1 G n (g) n (d) f(d)ĥ(d) m=0 q m (1 + m) 2 α

7 Beta distribution on arithmetical semigroups 153 Consider the remainder term in (36) By Lemma 22 r(n t) r 1 (n t) := 1 G n (g) m nt Further using (34) and Lemma 21 we deduce Finally (22) yields q n m (1 + n m) 2 α (d)=m r 1 (n t) (1 + n) 1 κ T n (t; 1 κ + α 2 α) (37) r(n t) r 1 (n t) 1 (ln n)ɛ(α) + nα n The main term in (36) by Lemma 22 becomes Φ n (t) = Aα L(α h) Γ(α)G n (g) The latter sum we can write m nt f(d)ĥ(d) f(d) h(d)q n (d) (1 + n (d)) 1 α + O (r 1(n t)) q n m (1 + n m) 1 α Therefore (34) and Lemma 21 imply (d)=m f(d) h(d) Φ n (t) = Γ(κ)L(α h)l(κ α f h) T n (t; 1 κ + α 1 α) Γ(α)Γ(κ α)l(κ g) (1 + n) κ ( ) Tn (t; 2 κ + α 1 α) O (1 + n) κ 1 + r 1 (n t) The routine calculation yields that L(α h)l(κ α f h) = L(κ g) (see eg [3]) Now we may apply (37) and Lemma 23 to obtain Γ(κ) Φ n (t) = Γ(α)Γ(κ α) J(t; 1 κ + α 1 α n 1 ) + ( 1 + O n κ α + 1 n α + (ln n)ɛ(κ α) + (ln n) ɛ(α) ) n This estimate (23) (36) and (35) complete the proof of Theorem The authors are thankful to the referee for valuable suggestions

8 154 G Bareikis and A Mačiulis References [1] Bareikis G and A Mačiulis On the numbers of divisors in arithmetical semigroup Ann Univ Sci Budapest Sect Comp 39 (2013) [2] Bareikis G and A Mačiulis Asymptotic expectation of a sequence of arithmetical processes Anal Probab Methods in Number Theory Kubilius Memorial Volume TEV Vilnius [3] Bareikis G and A Mačiulis Modeling the beta distribution using multiplicative functions Lith Math J 57(2) (2017) [4] Bareikis G and E Manstavičius On the DDT theorem Acta Arith 126(2007) [5] Daoud MS A Hidri and M Naimi The distribution law of divisors on a sequence of integers Lith Math J 55(4) (2015) [6] Deshouillers JM F Dress and G Tenenbaum Lois de répartition des diviseurs Acta Arith 34 (1979) 7 19 [7] Flajolet P and A Odlyzko Singularity analysis of generating functions SIAM J Discrete Math 3(2) (1990) [8] Indlekofer K-H E Manstavičius and R Warlimont On a certain class of infinite products with an application to arithmetical semigroups Archiv der Math 56 (1991) [9] Indlekofer K-H and E Manstavičius Additive and multiplicative functions on arithmetical semigroups Publ Math Debrecen 45 (1994) 1 17 [10] Knopfmacher J Analytic Arithmetic of Algebraic Function Fields Marcel Dekker New York 1979 G Bareikis and A Mačiulis Vilnius University Faculty of Mathematics and Informatics Institute of Computer Science Vilnius Lithuania gintautasbareikis@mifvult algirdasmaciulis@mifvult