THE DISTRIBUTION OF ADDITIVE FUNCTIONS IN SHORT INTERVALS ON THE SET OF SHIFTED INTEGERS HAVING A FIXED NUMBER OF PRIME FACTORS

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1 Annales Univ. Sci. Budaest., Sect. Com ) THE DISTRIBUTION OF ADDITIVE FUNCTIONS IN SHORT INTERVALS ON THE SET OF SHIFTED INTEGERS HAVING A FIXED NUMBER OF PRIME FACTORS J.-M. De Koninck Québec, Canada) I. Kátai Budaest, Hungary) Dedicated to Dr. Bui Minh Phong on his sixtieth birthday Communicated by K.-H. Indlekofer Received Setember 9, 202) Abstract. Given a strongly additive function f, we establish short interval estimates for f on the set of shifted rimes. We also consider similar sums, but running on sets of integers m +, where each integer m has a fixed number of rime factors.. Introduction Given integers q 2 and a 0, let ψx; q, a) = n x n a mod q) Λn), The first author suorted in art by a grant from NSERC, the second author suorted by ELTE IK.

2 58 J.-M. De Koninck and I. Kátai where Λn) stands for the von Mangoldt function. Let also φ stand for the Euler function. The well known Bombieri-Vinogradov theorem see Bombieri [3] and Vinogradov [6]) rovides an estimate for the error term in the Prime Number Theorem for arithmetic rogressions, averaged over the moduli q Q; it can be stated as follows. Bombieri-Vinogradov theorem. Given an arbitrary number A > 0, there exists B = BA) > 0 such that where Q = x log B x. q Q a,q)= ψx; q, a) x x φq) = O log A x The roblem of finding an estimate similar to the Bombieri-Vinogradov theorem for short intervals was first studied by Jutila [9] who obtained an estimate of the form.) q Q a q x h y a,q)= 2 z x ), ψz + h; q, a) ψz; q, a) h φq) y log A x, where, if we set y = x θ and Q = x η / log B x, the exonent η is bounded by a certain value which deends on θ and on { ) } inf ξ : ζ 2 + it t ξ, where ζ is the Riemann zeta function. This estimate was later imroved by various authors, namely Huxley & Iwaniec [8], Ricci [4], Perelli, Pintz & Salerno [2], [3], Zhan [7] and Timofeev [5]. Using the estimate obtained by Perelli, Pintz & Salerno [3], one can relace ψx; q, a) by πx; q, a) := #{ x : a mod q)} in order to obtain the following version of the Bombieri-Vinogradov theorem for short intervals..2) Theorem A. q Q a q a,q)= 2 h y x 2 z x lih) πz + h; q, a) πz; q, a) ϕq) y log A x,

3 The distribution of additive functions in short intervals 59 x where y = x 7 2 +ε, Q = x /40 dt and lix) :=. Here A > 0 and ε > 0 are log t arbitrary constants, with the imlied constants in deending only on A and ε. Recall the well known Erdős-Kac and Erdős-Wintner theorems. Erdős-Kac theorem. Let fn) be a strongly additive function and let Φz) := z e t2 /2 dt stand for the normal Gaussian distribution. Further 2π set Ax) := x f) 0 and Bx) := x and assume that Bx) as x. f 2 ) { } lim x x # fn) Ax) n x : z = Φz). Bx) The above result was established by Erdős and Kac in 939 [5]. Erdős-Wintner theorem. Let fn) be an additive function. f ossesses a distribution function if and only if each of the three series f) >, f) f), f) f 2 ) are convergent. This result was established by Erdős and Wintner in 939 [6]. 2. First series of main results Let ε > 0 be a fixed small number. Let πx) stand for the number of rime numbers not exceeding x. Let I x,y = [x, x + y], where x 7 2 +ε y x, and let πi x,y ) := I x,y. By using standard techniques, we can rove the following theorems.

4 60 J.-M. De Koninck and I. Kátai Theorem. Let g be a strongly multilicative function such that g) and g) as. Assume that the infinite sum g) converges. Letting Mg) := + g) ). x 7/2+ε y x g + ) Mg) πi x,y ) 0 asx. I x,y Theorem 2. Let f be a strongly additive function such that f) 0 for all rimes and such that f) 0 as. Let Ax) = f) and x assume that f 2 ) <. Moreover, let ϕτ) := ) + eiτf) e iτf)/ ) and let F u) be the distribution function whose characteristic function is ϕτ). Finally, let F Ix,y u) := πi x,y ) # { I x,y : f + ) Ax) < y}. lim x x 7/2+ε y x F Ix,y u) F u) = 0. u R Theorem 3. Let f be a strongly additive function and set Ax) = x and Bx) = x x. lim x x 7/2+ε y x u R f) f 2 ) f). Assume that Bx) and that x Bx) 0 as πi x,y ) # { } f + ) Ax) I x,y : < u Bx) Φu) = 0.

5 The distribution of additive functions in short intervals 6 The second author roved [0] that if g is a multilicative function satisfying gn) for all n, and 2.) P g) is convergent, and if Ng) is the roduct Ng) = + k= ) g k ) k, then 2.2) lim g + ) = Ng). x πx) x Hence, he deduced that if f is additive and satisfies the 3-series conditions.3), then the limit lim x #{ x : f + ) < z} = F z) πx) exists for almost all z R, meaning in other words that f has a limiting distribution on the set of shifted rimes. In the roof of this result, the Bombieri-Vinogradov inequality was used. However, the full strength of the inequality was not necessary. In fact, the inequality due to Barban [] was sufficient, namely the following: For a certain constant δ > 0 and every fixed A > 0, l,k)= k x δ µ 2 k) In fact, any ositive δ < 3 23 lix) πx; k, l) φk) < x log A x. is admissible. Now, the natural question is can we deduce from Theorem A a short interval version of the Erdős-Wintner theorem for shifted rimes or not? In fact, one could easily construct a strongly additive function f which is 0 on a set of rimes 0 such that 0 / < and such that f) {, } for all \ 0, while the short interval version of the Erdős-Wintner theorem for shifted rimes does not hold.

6 62 J.-M. De Koninck and I. Kátai Let us now assume that the condition 2.) is comlemented by the fact that g) as. the short interval version of 2.2) can be roved by the method of the second author alying Theorem A see Theorem ). Hence, we can deduce the following assertion: Let f is an additive function such that f) 0 as. assume that both series Further f) and f 2 ) converge. the function F Ix,y u) := πi x,y ) #{ I x,y : f + ) < u} has a limit distribution F u). Moreover, the characteristic function of F is given by ϕ F τ) = P + k= e iτfk ) Theorem 4. Let f be a strongly additive function such that f) for all rimes. Let h Z[x]. For each integer d, let ηd) denote the number of residue classes r mod d) which are corime with d and which satisfy hr) 0 mod d). Let also 2.3) Ax) = x η) f) k ) and Bx) = η) f 2 ). x. Assume that Bx) as x. lim x x 7/2+ε y x u R { πi x,y ) # I x,y : f h) ) Ax) Bx) } < u Φu) = 0. Theorem 5. Let f be a strongly additive function such that f) 0 as. Let h and η be as in Theorem 4. Assume also that the two series η) f) and η) f 2 )

7 The distribution of additive functions in short intervals 63 are convergent. It is known that the limit distribution F z) := lim #{ x : f h) ) < z} x πx) exists see Theorem 2.4 in the book of Elliott [4]). lim x x 7/2+ε y x z R πi x,y ) # { I x,y : f h) ) < z} F z) = 0. Since the roofs of Theorems -5 can be obtained on the same way as their non short versions, we shall omit them. Remark. The condition f) 0 as in Theorems, 2 and 5 and the condition Bx) x f) 0 as x in Theorems 3 and 4 allows us to evaluate f and g. We can rove a Turan-Kubilius tye inequality and roceed in the usual way. Remark 2. Observe that it can be shown that the above theorems remain true if we consider the values over the set a + with I x,y, where a x ε/2, say. Given an integer n 2, let ωn) stand for the number of distinct rime factors of n and Ωn) for the number of rime factors of n counting their multilicity, and further set ω) = Ω) = 0. We now define In Kátai [], it was roved that k := {n N : ωn) = k}, N k := {n N : Ωn) = k}, Π k I x,y ) := #{n I x,y : ωn) = k}, N k I x,y ) := #{n I x,y : Ωn) = k}. y Π k I x,y ) = + o)) log x log log x) k k )! x ) uniformly for ositive integers k log log x+c x log log x, where cx is a function which tends to infinity, but sufficiently slowly. Using essentially the same method, it was later roved by Bassily and Kátai [2] that, in the same range of k, y log log x) k N k I x,y ) = + o)) x ). log x k )!

8 64 J.-M. De Koninck and I. Kátai It is highly robable that the analogues of Theorems through 5 hold if we relace the set of rimes by the set of integers in k uniformly for k < < B log log x for any arbitrary fixed number B or by the set of integers n N k uniformly for k < 2 ε) log log x for any arbitrary small number ε > 0. Such a result would follow if we could rove the analogue of the short interval version of the Bombieri-Vinogradov theorem as in Theorem A, substituting by m k or m N k. But as of today, we cannot rove this. Nevertheless, we can rove that the analogues of Theorems -5 hold uniformly for m k and m N k uniformly for k k x, rovided k 2 x/ log log x 0 as x. To rove this, we need the following lemma, where P n) res. P 2 n)) stands for the largest res. second largest) rime factor of the integer n 2, with P ) = P 2 ) =. Lemma. Let 2 k x N be such that ρ x := k 2 x/ log log x 0 as x. Set θ x := ρ x and let 2.4) Similarly, 2.5) Π 0) k I x,y) = #{n P k I x,y : P 2 n) x θx/2k }, N 0) k I x,y) = #{n N k I x,y : P 2 n) x θ x/2k }. Π 0) k I x,y) 2 k k x Π k I x,y ) N 0) k I x,y) 2 k k x N k I x,y ) 0 as x. 0 as x. Proof. We first rove 2.4). Let δ = θ x /2k. Consider the integers n = = k I x,y with < < k and let m be the largest of those rime factors satisfying m < x δ. If n is counted in Π 0) k I x,y), then m k 2 and n = m ν = aν, say, where P ν) > x δ. We then have a x δm < x /2. Hence, for fixed,..., m, the number of such ν s is, by Mertens theorem, <# ν : ν, cy a π<x δ π x δ π ) π =, c y aδ log x. x a ν x a + y a

9 The distribution of additive functions in short intervals 65 Summing over m = 0 that is, when a = ) and m =,..., k and observing that, for fixed m, we obtain that Since k a=0 ωa)=m a < m )! log log x + c)m, Π 0) k I x,y) c k 2 2y log log x + c) m δ log x m! m=0 k log log x δ c 3y log log x + c) k 2 c 4 Π k I x,y ) k δ log x k 2)! log log x δ. 2k2 x log log x estimate 2.4) follows immediately. θ x = 2 ρ x 0 as x, The roof of 2.5) is similar and will therefore be omitted. 3. Second series of main results We can rove the following generalizations of Theorems through 5. Theorem 6. Let g be as in the statement of Theorem and let M a g) := + g) ) for a x ε. a x ε/2 x 7/2+ε y x,a)= ga + ) M a g) πi x,y ) 0 as x. I x,y Theorem 7. Let f, ϕ and F be as in Theorem 2 and let Ax) be as in Theorem 2. Moreover, let F k) I x,y u) := Π k I x,y ) #{m I x,y k : fm + ) Ax) < u}.

10 66 J.-M. De Koninck and I. Kátai lim su x k k x x 7/2+ε y x u R F k) I x,y u) F u) = 0. Theorem 8. Let f, Ax) and Bx) be as in Theorem 3, with Bx) as x. Moreover, let { } G k) I x,y u) := Π k I x,y ) # fm + ) Ax) m I x,y k : < u. Bx) lim su x su G k) k k x x 7/2+ε y x u R I x,y u) Φu) = 0. Theorem 9. Let f, h, η, Ax) and Bx) be as in Theorem 4, with Bx) as x. Moreover, let { } H k) I x,y u) := Π k I x,y ) # f hm) ) Ax) m I x,y k : < u. Bx) lim su x su H k) k k x x 7/2+ε y x u R I x,y u) Φu) = 0. Theorem 0. Let f, h and η be as in Theorem 5. Moreover, assume that the two series η) f) and η) f 2 ) are convergent, and let lim x x 7/2+ε y x z R F z) := lim #{ x : f h) ) < z}. x πx) π k I x,y ) # {m I x,y k : f hm) ) < z} F z) = 0. Remark 3. Using the result of Germán [7], one can rove the analogue of Theorems 6 and 7 for the non short interval case.

11 The distribution of additive functions in short intervals Proof of Theorem 6 Since the roof of Theorem 6 is essentially a model for the roofs of Theorems 7-0, we will only rove Theorem 6. Let K x) < K 2 x) be two numbers such that lim x K x) = and set K 2 x) = x δ, where δ > 0 is a small number. For each number H > 0, set g) if H, g H ) = if > H. Define imlicitly the strongly additive function f by g) = ex{if)} for f) [ π, π). Further define f) if H, f H ) = 0 if > H. In light of the condition lim g) =, we obtain that gn) g K2 x)n) 0 as x. n I x,y Let x be a large number with corresonding numbers K < K 2. Finally, set un) = f). n K <K 2 Using.2), we can obtain a Turán-Kubilius tye inequality. Indeed, letting A a := [K,K 2 ] /a f), B2 a = [K,K 2 ] /a f 2 ), we have Now I x,y ua + ) A a ) 2 cπi x,y )B 2 a. A a = A K,K 2 ] a f) = A D a,

12 68 J.-M. De Koninck and I. Kátai say. From the conditions stated in the theorem, we obtain that B a 0 and A a 0 uniformly for a x ε as x. We have thus obtained that, uniformly for a [, x ε ], πi x,y ) I x,y ga + ) g K a + )e ida 0 as x. Now, let g K n) = h K d). Since g is strongly multilicative, it follows d n that h α ) = 0 if α 2 and also that h K ) = 0 if > K. Let us now choose K = δ log x, where δ is a small ositive number. if h K d) 0, we then have that On the other hand, 4.) d π e 2K = x 2δ, so that d x 2δ. π K I x,y g K a + ) = d,a)= h K d)πi x,y d, l d ), where l d is the solution of al d + 0 mod d). Since g) as, it follows that h) 0 as. Thus, h K d) is bounded. Hence, from.2) and 4.), we have 4.2) I x,y g K a+) = d,a)= h K d) φd) πi x,y)+oπi x,y )) = E a πi x,y )+oπi x,y )), say. But it is clear that E a = K /a + g) ). Using this last estimate in 4.2) and recalling the definition of D a, we obtain that πi x,y ) I x,y ga + ) = e ida K /a + g) ) + o) x ).

13 The distribution of additive functions in short intervals 69 Now, observe that >K a + g) ) = ) + eif) = >K a = >K a + if) f 2 )) + O ) ) 2 = = e ida + o)). Since >K + g) ) as x, the roof of Theorem 6 is comlete. References [] Barban, M.B., New alications of the large sieve of Yu.V. Linnik, Akad. Nauk. Uzk. SSR Trudy Inst. Mat. V.I. Romanov, Teor. Ver. Mat. Stat., 22 96), -20. in Russian) [2] Bassily, N.L. and Kátai, I., Some further remarks on a aer of K. Ramachandra, Annales Univ. Sci. Budaest. Sect. Com., 34 20), [3] Bombieri, E., On the large sieve, Mathematika, 2 965), [4] Elliott, P.D.T.A., Probabilistic number theory II: Central limit theorems, Sringer-Verlag, 980. [5] Erdős, P. and Kac, M., On the Gaussian law of errors in the theory of additive functions, Proc. Nat. Acad. Sci. USA, ), [6] Erdős, P. and Wintner, A., Additive arithmetical functions and statistical indeendence, Amer. J. Math., 6 939), [7] 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., ), [8] Huxley, M.N. and Iwaniec, H., Bombieri s theorem in short intervals, Mathematika, ), [9] Jutila, M., A statistical density theorem for L-functions with alications, Acta Arith., 6 969),

14 70 J.-M. De Koninck and I. Kátai [0] Kátai, I., On the distribution of arithmetical functions on the set of rime lus one, Comositio Math., 9 968), [] Kátai, I., A remark on a aer of K. Ramachandra, Number theory Ootacamund, 984), Lecture Notes in Math. 22, Sringer, Berlin, 985, [2] Perelli, A., Pintz, J. and Salerno, S., Bombieri s theorem in short intervals, Ann. Scvola Normale Su. Pisa, Serie IV, 984), [3] Perelli, A., Pintz, J. and Salerno, S., Bombieri s theorem in short intervals II., Invent. Math., ), -9. [4] Ricci, S.J., Mean-value theorems for rimes in short intervals, Proc. London Math. Soc. 37 2) 978), [5] Timofeev, N.M., Distribution of arithmetic functions in short intervals in the mean with resect to rogressions, Izv. Akad. Nauk. SSSR Ser. Mat., 5 987), , 447. in Russian) [6] Vinogradov, A.I., The density conjecture for Dirichlet L-series, Izv. Akad. Nauk. SSSR Ser. Mat., ), [7] Zhan, T., Bombieri s theorem in short intervals, Acta Math. Sinica, 5 989), J.-M. De Koninck Déartement de mathématiques et de statistique Université Laval Québec Québec GV 0A6, Canada jmdk@mat.ulaval.ca I. Kátai Deartment of Comuter Algebra Eötvös Loránd University Pázmány Péter s. /C H-7 Budaest, Hungary katai@comalg.inf.elte.hu