Eötvös Loránd University Faculty of Informatics. Distribution of additive arithmetical functions

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1 Eötvös Loránd University Faculty of Informatics Distribution of additive arithmetical functions Theses of Ph.D. Dissertation by László Germán Suervisor Prof. Dr. Imre Kátai member of the Hungarian Academy of Sciences Doctoral School of Informatics Director: Prof. Dr. János Demetrovics member of the Hungarian Academy of Sciences Doctoral Program: Numerical and Symbolical Comutation Program Director: Prof. Dr. Antal Járai Doctor of Mathematical Sciences 2008.

2 Introduction - aim of the work An arithmetical function f : N C is said to be additive if f(n m) = f(n) + f(m) whenever (m, n) =. Such a function is comletely determined by its values on rime owers. If f(n m) = f(n)+f(m) holds unconditionally, then f is comletely additive, and if f( α ) = f() is constant for α, whenever is a rime number, then f is said to be strongly additive. Let f be a real additive function, and 0 < x. The frequency of f is defined by F x (z) := x n x f(n) z for all real z. Then F x (z) is a distribution function and we say that f has a limit distribution, if, F x (z) F (z) (x ), in all z continuity oints of F, where F is a distribution function as well. This tye of convergence has been called weak convergence and we will denote it by F x F (x ). It turned out, that the above roerty of f is equivalent to the following condition, which we call the Erdős-Wintner condition, i.e. the three series over rime numbers f() >, f() f(), f() f 2 () converge. Let A x be a subset of N such that A x [..x] is nonemty, with < x. Then the frequency of f on A x is now defined by ν x (n A x ; f(n) z) := A x [..x] n x n Ax f(n) z Then one can ask if the frequencies ossess a limit law or not, that is if () ν x (n A x ; f(n) z) F (z) (x ).

3 or not with a distribution function F (z). Erdős and Wintner considered this question taking A x to be N (See [2] for examle). Kátai and Hildebrand ([4], [3]) investigated the case A x = P +, where P denotes the set of rimes. The main toic of the thesis is to investigate this tye of distribution roblems if we take A x = {n x : ω(n ) = k x }, where ω(n) denotes the number of distinct rime factors of n, and k x ε(x) log log x with ε(x) 0 (x ). Note that the roblem of Kátai and Hildebrand corresonds to the case k x = (aart from the higher rime owers, which have zero relative density in the set of rime owers). An alication of the results discussed in Chater 3. allows us to give an Erdős-Kac tye Theorem in Chater 6. which seems to be stated in the literature only for fixed k, and with further strong restrictions on the set A x (See []). Further notations in this summary: the set of ositive integers having k distinct rime factors is denoted by P k. The set of numbers in P k not larger than x is denoted by P k (x). The number of integers in P k (x) is denoted by π k (x). If k = we suress the suffixes. 2 Alied methods There are several methods to investigate distribution roblems arising from number theory. One of the most imortant is the model of Kubilius, is discussed in Chater 3. The continuity theorem of Lévy allows us to transfer questions concerning weak convergence of distribution functions into mean value estimations for multilicative functions of modulus one. That is, () holds in all z continuity oints of F if and only if lim x A x [..x] n x n Ax e itf(n) exists for all real t, and the function ψ(t) defined by the value of this limit is continuous in t = 0. In this case the characteristic function of F equals to ψ of course. This method was used in Chater 4. Although the necessary mean value estimations are done here, in Chater 5. there are more general results. The method used here, which is resent in Chater 4, and Chater 6 also allows us to transfer the roblem from the set P k to the set of shifted rimes. 2

4 In Chater 6. the method of Chater. is used in the combination with the method of Chater 4. In Chater 4. we rove the necessity of the Erdős-Wintner condition using the method of Hildebrand (See [3]). 3 Results 3. Results of Chater 2. In this Chater we reare the necessary results concerning the structure of the set P k. A relevant condition we will work with is the following: let ε(x) be a function such that ε(x) 0 as x. We say that k has roerty A(ε, x) if 2 k ε(x) log log x. Suose that n is in the form Let n = r r 2 2 r k k < 2 < < k. δ j (n) = r r j j (j =,..., k), and γ j (n) = log j+ log δ j (n) (j =,..., k ). The first result we rove, shows that for almost all integers in P k the size of the j + -th rime factor is large in terms of the roduct of the first j rime factor of n. That is, if then we have the following (n) = min γ j(n) j=,...k Lemma. Suose k has roerty A(ε, x) and let M = M x be defined by log log M = log log x. Then there exists a sequence A x, deending at most on ε(x) with A x as x such that and (π k ) A x, r = r 2 = = r k =, > M holds for all but o()π k (x) element of P k (x) as x. 3

5 3.2 Results of Chater 3. The main result of this chater describes the robabilistic nature of additive functions on the set of shifted rimes. Lemma 2. Let f(n) be a real strongly additive arithmetical function, x > 2, 0 < σ and let for some r x σ. Let f r (n) = n r f(), K D (x) = {D + x : P}. Then with an arbitrary constant A > 0 we have (ν x =)ν(n K D (x) : f r (n) z) = P( r D X z) + O(ex( log x log r ) + log A x) uniformly for all D x σ, where for all rimes the X s are indeendent random variables over some aroriate robability sace (Ω, A, P) defined by their distribution as { f() with robability ϕ() X = 0 with robability. ϕ() 3.3 Results of Chater 4. The corresonding Erdős-Wintner theorem is as follows: Theorem. Let f be a real additive function. Let F k,x (z) := ν x (n P k + ; f(n) z). Assume that there is a sequence k = k x, for which all the terms have the roerty A(ε, x), and a distribution function F such that F k,x F. Then the Erdős-Wintner condition holds. Conversely, assume that the Erdős-Wintner condition holds for f. Then with a distribution function G we have max 2 k ε(x) F k,x (z) G(z) 0 (x ) log log x 4

6 for all z continuity oints of G. Consequently F = G. The characteristic function of F is given by ϕ(t) = ( + h()), where h () = + m= e itf(m ) m. To rove this result we use a generalization of the result of Kátai for DP +, which is the following Theorem 2. With the notations of the above theorem, let f be a real additive function, and assume that f() >, f() f 2 () are convergent. Let σ > 0, and ϱ = min{σ/4, /4}. Let A (x) := f (), a (m) := f() x f() m Let K D (x) = {D + x P}. f (). F D,x (y) := ( (( ) ϱ ) x ν x (n K D (x), f (n) A D ) ) a (D) z, with characteristic function ϕ D,x (t), and let ϕ D (t) := f() it ( + h ()) ( + h ()) e. Then f() > D f() D max ϕ D,x (t) ϕ D (t) 0 D x σ uniformly for all bounded values of t, i.e. constant. 5 (x ), if t < T, T is an arbitrary

7 3.4 Results of Chater 5. One is able to give general mean value estimations in the following form: Theorem 3. Let f(n) be a multilicative function of modulus. Let furthermore d be a ositive integer. Suose that there is a real τ such that the series χ()f() iτ 2 converge for some rimitive character χ (mod d). Then ( ( )) x π g(d + ) = xiτ µ(d) D + iτ ϕ(d) D+ x ( x dd + α + o() (x ) uniformly for all D x ε with (d, D) =, where 0 ε <. ) f( α ) iατ χ( α ) α As an alication of Theorem 3 we are able to analyze the mean behavior of multilicative functions on the set P k + in some cases. Theorem 4. Let g(n) be a multilicative function of modulus one, such that there is a rimitive character χ (mod d) for some fixed d and a real τ such that converges. Then π k (x) n x ω(n)=k g(n + ) = + iτ Reχ()g() iτ xiτ µ(d) ϕ(d) ( x d + o() (x ) uniformly for all k having roerty A(ε, x). + α ) f( α ) iατ χ( α ) α 6

8 3.5 Results of Chater 6. In this section we give the relevant Erdős-Kac tye results. The distribution function G(z) stands for the Gaussian distribution. Theorem 5. Let f(m) be a real additive function satisfying B 2 (x) for all fixed ε > 0 where x f() >εb(x) B(x) = ( x f 2 () 0 f 2 () )/2. (x ), Then we have with A(x) = x f() that ν x (n P k (x) : f(n + ) A(x) B(x) z) G(z) (x ) uniformly for all k having roerty A(ε, x). With the notations of the above theorem, we say that f(n) belongs to the class H if there exists a function r = r(x) such that log r log x 0, B(r) B(x), B(x) as x. This class of functions was introduced by Kubilius. We roceed similarly as in the revious chaters. First we rove the following Theorem 6. Let f(m) be an additive function of class H. Let B D (x) = ( x D f 2 () )/2, and let δ(x) be a sequence tending to zero arbitrary slowly. Furthermore let 0 < σ and let (T δ (x) =) T (x) = {D x σ : B D (x) = B(x)( + O(δ(x)))}. 7

9 Then we have with A D (x) = x D f() that ν x (n K D (x) : f(n) A D(x) B D (x) z) G(z) (x ) uniformly for all ositive integer D T (x), if and only if for each fixed ε > 0 B 2 D (x) x D f() >εb D (x) f 2 () 0 (x ) uniformly for all D T (x). A direct consequence of Theorem 5 is the following Corollary. ν x (n x, ω(n) = k : ω(n + ) log log x log log x z) 2π z e w2 /2 dw (x ) uniformly for all k having roerty A(ε, x). References [] M. B. Barban, The Large Sieve method and its alications in the theory of numbers, Russ. Math. Surv. 2 (966), [2] P. D. T. A. Elliott, Probabilistic Number Theory I., Sringer-Verlag, New York, 979. [3] A. Hildebrand, Additive and multilicative functions on shifted rimes, Proc. London Math. Soc. 53 (989), [4] I. Kátai, On the distribution of arithmetical functions on the set of rimes lus one, Comosito Math. 9 (968),

10 Publications of the author [] Germán, L. and Kovács, A., On number system constructions, Acta Math. Hungar. 5 (2007), no. -2, 55 67; [2] Germán, L. and Kátai, I., Distribution of q-additive functions on the set of integers having k rime factors, Annales Univ. Sci. Budaest. Sect. Com., 27 (2007), [3] Germán, L., The distribution of an additive arithmetical function on the set of shifted integers having k distinct rime factors, Annales Univ. Sci. Budaest. Sect. Com., 27 (2007), Conference talks [] Germán, L., The distribution of an additive arithmetical function on the set of shifted integers having k distinct rime factors, The IVth international conference on analytic and robabilistic methods in number theory, 9/2006, Palanga [2] Germán, L., On multilicative functions on consecutive integers, Conference talk Numbers, Functions, Equations 08, 06/2008, Noszvaj 9