Annales Univ. Sci. Budapest., Sect. Comp. 43 204 253 265 SOME RESULTS AND PROBLEMS IN PROBABILISTIC NUMBER THEORY Imre Kátai and Bui Minh Phong Budapest, Hungary Le Manh Thanh Hue, Vietnam Communicated by László Germán Received August 5, 203; accepted February, 204 Abstract. We formulate some results, open problems and conjectures in probabilistic number theory.. Introduction Notation. Let, as usual, P, N, N 0, Z, R, C be the set of primes, positive integers, non-negative integers, integers, real and complex numbers, respectively. We say that f : N R is an additive function, if f = 0 and fmn = fm + fn for all m, n =. Let A denote set of all additive functions. A function g : N C is multiplicative, if g = and gmn = gm gn for all m, n =. We denote by M the set of all multiplicative functions. Key words and phrases: multiplicative functions, additive functions, continuous homomorphism. 200 Mathematics Subject Classification: A07, A25, N25, N64. This work was completed with the support of the Hungarian and Vietnamese TET grant agreement no. TET 0--20-0645.
254 Kátai, I., Bui Minh Phong and Le Manh Thanh Let q 2, integer, A q = {0,..., q }. Every n N 0 can be written as n = k ε j nq j, ε k n 0, ε j n A q, and this expansion is unique. In the following let A q denote the set of q-additive functions, i.e. a function k f : N 0 R belongs to A q, if f0 = 0, and fn = fε j nq j n N 0. Similarly, we say that g : N 0 C is q-multiplicative, if g0 =, and gn = k gε j nq j n N. Let M q be the set of q-multiplicative functions. Let πx be the number of the primes up to x, and πx, k, l be the number of the primes p x satisfying p l mod k. Examples. ωn= number of prime factors of n, ωn A, τn = number of divisors of n, τn M, cm fn = log n A, cm φn = Euler s totient function, φn M, cm σn = sum of divisors of n, σn M. If fn A, gn = z fn, z C, then g M. If αn = k ε j n = sum of digits functions, then αn A q. fn = n A q, gn = z n M q. The following wellknown results can be found in [8]. Definition. We say that f A has a it distribution on the set N if has a it F. F N y := #{n N fn < y} N F Ny = F y for almost all y or in all continuity points of N
Some results and problems in probabilistic number theory 255 Theorem Erdős-Wintner. f A has a it distribution if and only if the next three series converge. fp > p,.2 fp fp p,.3 fp f 2 p. p Theorem 2. f A has a it distribution with a suitable centralization cn, i.e..4 N #{n N fn cn < y} = F y N exists for almost all y, if and only if.,.3 converge. Let an := and α is finite. fp p<n Theorem 3 Erdős-Kac. Let f A,.5 A N := p N Assume that fp = O N Φ = Gaussian law. N # fp. If.4 holds, then an cn = α exists, p fp p, B2 N = p P. Then { n N p α N fn AN B N f 2 p α p α. } < y = Φy, Turán-Kubilius inequality. Let f A. Then fn A N 2 cnbn 2, n N A N, B N are defined in.5.
256 Kátai, I., Bui Minh Phong and Le Manh Thanh 2. Distribution of additive functions on some subsets of integers Let B N, Bx = #{n x, n B}. We say that f A is distributed in it with centralization a N and normalization b N on the set B, if { } N BN # n N, n B fn a N < y = F y b N exists a.a. Results:. B = P + = {p+ p P} = set of shifted primes, a N = 0, b N = 0. f A has a it distribution on P + if the 3 series in Erdős-Wintner theorem are convergent [6]. The proof is based on the method of characteristic functions: gn = e iτfn. 2. gp + = M τ x = x πx p<x = p + gp p + gp2 p 2 +. p To prove 2., the Siegel-Walfisz theorem was used, namely that πx, k, l = = li x + O e c log x φk p x p l mod k uniformly as l, k =, k < log x c, where li x = x 2 du log u, and the A. I. Vinogradov - E. Bombieri inequality, according to it max max li y l,k= y x πy, k, l φk C x log x B, k x log x A
Some results and problems in probabilistic number theory 257 B = 2A + 5, C is a suitable non-effective constant. J. Kubilius and P. Erdős asked on the necessity of the convergence of the 3 series.,.2,.3. Partial results were obtained by P.D.T.A. Elliott, I. Kátai, and N. Timofeev. Finally more than 20 years later it was proved by A. Hidebrand that they are necessary [4]. 2. L. Germán proved the analog of Erdős-Wintner theorem on the set where εn 0. See [2]. B = {n + ωn εn log log n}, 3. On q-additive functions Let f A q, ξ j be independent random variables, P ξ j = faq j = q a A q, η N = ξ 0 + ξ +... + ξ N, m j = Eξ j = faq j, q a A q σj 2 = f 2 aq j m 2 q j. a A q By using the standard method of probability theory one can prove Theorem 4. Let f A q. Then x #{n < x fn < y} = F y x for almost all y exists, if and only if 3., 3.2 are convergent: 3. a A q faq j, 3.2 a A q f 2 aq j.
258 Kátai, I., Bui Minh Phong and Le Manh Thanh What is happening if we consider the distribution of f over a subset B in N 0? Let j <... < j h < N, b,..., b h A q, j,..., j h B N b,..., b h B N j,..., j h b,..., b h = {m < q N m B, ε jl m = b l, l =,..., h}, = #{m < q N m B, ε jl m = b l, l =,..., h}, Bx = #{m < x m B}. Let P u Z[u] be a polynomial r = deg P, P u = a r u r +... + a u + a 0, a r > 0. Let l Σ := A x,..., l h = #{n x ε lj P n = b j, j =,..., h}, b,..., b h Σ 2 := l x,..., l h = #{p x ε lj P p = b j, j =,..., h}. b,..., b h Lemma. Let q N x < q N+. Let h be fixed, λ be an arbitrary constant, 3.3 Then N 3 l <... < l h rn N 3. Σ = x q h + O x log x λ, Σ 2 = πx q h + O x log x λ uniformly as l,..., l h in 3.3, b,..., b h A q. The proof depends on the next Lemmas 2 and 3. Let eu := e 2πiu. Lemma 2 Hua Loo Keng. Let 0 < Q c klog x τ S = p x p t mod Q efp and in which fy = h q yk + α y k + + a k, h, q =.
Some results and problems in probabilistic number theory 259 Suppose that log x τ q x k log x τ. For arbitrary τ 0 > 0, when τ > 2 6k τ 0 + τ +, we always have c 2 k depends only on k. S c 2 kx log x τ0 Q. Lemma 3 I. M. Vinogradov. Let f be as in Lemma 2, S = n x efn. Let τ 0, τ 3, τ 4 be arbitrary positive numbers, Suppose Then τ 2 k τ 0 + τ 3 + 2kτ 4 + 2 3k 2. log x τ < q x k log x τ. S xlog x τ. The constant standing implicitly in may depend on τ 3, τ. Theorem 5. [] Let f A q, fbq j = O as j, b E. Furthermore let. Here Dx log x 3 D 2 x = N σj 2 q N x < q N+. Let P u Z[u], deg P = r, P u u. Then {n x # < x fp n Mx r } 3.4 Dx r < y = Φy, 3.5 { πx # n < x fp p Mx r Dx r } < y = Φy, c k x = f n M x r k x r D x r. n x r From Lemma one can deduce that a k x = b k x = c k x.
260 Kátai, I., Bui Minh Phong and Le Manh Thanh It is clear that 3.4 is true if P u = u, furthermore c kx = u k dϕu = µ k. x Consequently a k x = µ k, b k x = µ k, and the Frechet-Shohat theorem implies the fulfilment of 3.4 and 3.5. How to prove Lemma? Let Let φ b x + n = φ b x Let b,..., b h A q, Let ty = F Then φ b x = n Z. { [ if x b q, b+ q, 0 otherwise in [0,. l <... < l h be arbitrary integers. Then F x,..., x h = φ b x... φ bh x h, { } [ n b ε j n = b q j+ q, b + q y q l +,..., tm = y q l h+ Let 0 < < 2q, f bx = 2.. { if ε lj m = b j, j =,..., h, 0 otherwise. ty = f b φ b x + udu, y q l + y... f bh q l. h+ We can prove that is small, furthermore tp n tp n n x ty = M T M emv y, M = [m,..., m h ], V = TM <. [ ] q l +,..., q l, h+
Some results and problems in probabilistic number theory 26 Thus tp n = T M n x tp p = T M p x AM e P n, H M AM e P p, H M V M = A M HM, H M, A M =. We can apply Lemma and 2. This completes the proof. Theorem 6. Let f A q. Assume that Mx exists and is finite, D 2 x <. Then there exist suitable distribution functions F y, F 2 y such that } {n x # 3.6 < x fp n < y = F y a.a. y, 3.7 { πx # p < x } fp p < y = F 2 y a.a. y. Theorem 7. If P n = n, and 3.7 holds true, then Mx exists and is finite, furthermore Dx is finite. x 4. Linear combinations of q - additive functions Here we mention some theorems without proof. In this section λ is the Lebesgue measure defined in R. Let f,..., f k A q ; a <... < a k < q, a i, q =, a i, a j = if i j, ln = f a n +... + f k a k n. Definition 2. We say that ln is a tight sequence if A N such that as K. CK := sup q N #{n < qn ln A N > K} 0 Theorem 8. K.-H. Indlekofer and I. Kátai, [5]
262 Kátai, I., Bui Minh Phong and Le Manh Thanh. ln is tight if and only if γ,..., γ k for which a γ +... + a k γ k = 0 and for ψ l n := f l n γ l n we have 4. l q ψ l aq j 2 < a=0 l =,..., k. 2. If the conditions of hold, then q N #{n < qn ln E N < y} = F y a.a. y, where F is a suitable distribution function, A l N := q N E N = a A q ψ l aq j, k l= A l N. 3. ln has a it distribution, if and only if the conditions of are satisfied, and if E N has a finite it. Theorem 9.. The sequence lp p P is tight if it is tight on the whole set N. 2. If 4. l hold, then N πq N #{p < qn lp E N < y} = F y a.a. y. 3. lp has a it distribution if and only if lp is tight and if E N has a finite it. Let µ l u = f l cq u, pu = q c A q N F N := pu, u=0 k µ l u, l= π u c,..., c k := q u+ #{n < qu+ ε u a j n = c j },
Some results and problems in probabilistic number theory 263 { [ cj πc,..., c k = λ x [0, ] {a j x} We have π u c,..., c k = πc,..., c k + O Let τ u := q, c j + q q u }, j =,..., k. u. c,...,c k A q f c q u +... + f k c k q u pu 2 π u c,..., c k, Theorem 0. Assume that for l =,..., k. Then N ϱ 2 N = N u=0 τ u. f l cq M ϱ M 0 as M q N # Theorem. Assume that and that as K N = [log N], N { n < q N ln F N ϱ N sup max M max c A q l f l cq M <, } < y = ϕy. ϱ 2 K N /ϱ 2 N 0, ϱ 2 N ϱ 2 N K N /ϱ 2 N 0 N. Then πq N # { p < q N lp F N ϱ N } < y = ϕy. 5. On a theorem of G. Harman and I. Kátai [3] Let 0 j < j 2 <... < j r N, x > expq 2, q N x q N. Let b = b,..., b r, j = j,..., j r, j x := #{p x ε lj p = b j, j =,..., r}, b j A x := #{n x ε lj n = b j, j =,..., r}. b
264 Kátai, I., Bui Minh Phong and Le Manh Thanh Theorem 2. Suppose that r < C N/ log N. Then j x = qr fb, j b log x A j x + O b x log log x φqq r log x 2, where q r if j > 0, fb, j = 0 if j = 0, b, q >, q r φq if j = 0, b, q =. A similar theorem is claimed in the paper [7] of I. Kátai, but there were some mistakes in the proof. Our guess: Theorem 2 remains valid up to r < 3 N. References [] Bassily, N.L. and I. Kátai, Distribution of the values of q-additive functions on polynomial sequences, Acta Math. Hungar., 68 4 995, 353 36. [2] Germán, L., The distribution of an additive arithmetical function on the set of shifted integers having k distinct prime factors, Annales Univ. Sci. Budapest. Sect. Comp., 27 2007, 87 25. [3] Harman, G. and I. Kátai, Primes with preassigned digits II., Acta Arith., 33 2 2008, 7-84. [4] Hildebrand, A., Additive and multiplicative functions on shifted primes, Proc. London Math. Soc., 53 989, 209-232. [5] Indlekofer, K.-H. and I. Kátai, Investigations in the theory of q- additive and q-multiplicative functions II., Acta Math. Hungar., 97 2002, 97-08 [6] Kátai, I., On distribution of arithmetical functions on the set of prime plus one, Compositio Math., 9 968, 278-289.
Some results and problems in probabilistic number theory 265 [7] Kátai, I., Distribution of digits of primes in q-ary canonical form, Acta. Math. Hungar., 47 3-4 986, 34-359. [8] Kubilius, J., Probabilistic Methods in the Theory of Numbers, Amer. Math. Soc. Translations of Math. Monographs, Providence, 964. Imre Kátai and Bui Minh Phong Le Manh Thanh Eötvös Loránd University Hue University H-7 Budapest Hue City Pázmány Péter Sétány I/C, 3 Le Loi Hungary Vietnam katai@compalg.inf.elte.hu lmthanh@hueuni.edu.vn bui@compalg.inf.elte.hu