Imre Kátai

Jean-Marie De Koninck

Publications Mathématiques de Besançon

Publications mathématiques de BesançonA l g è b r e  e t  t h é o r i e  d e s  n o m b r e sJean-Marie De Koninck and Imre KátaiThe number of large prime factors of integers and normal numbers2015, p. 5-12.<http://pmb.cedram.org/item?id=PMB_2015____5_0>© Presses universitaires de Franche-Comté, 2015, tous droits réservés.L’accès aux articles de la revue « Publications mathématiques de Besançon » (http://pmb.cedram.org/),implique l’accord avec les conditions générales d’utilisation (http://pmb.cedram.org/legal/). Toute utili-sation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ouimpression de ce fichier doit contenir la présente mention de copyright.Publication éditée par le laboratoire de mathématiquesde Besançon, UMR 6623 CNRS/UFCcedramArticle mis en ligne dans le cadre duCentre de diffusion des revues académiques de mathématiqueshttp://www.cedram.org/Publications mathématiques de Besançon – 2015, 5-12THE NUMBER OF LARGE PRIME FACTORS OF INTEGERS ANDNORMAL NUMBERSbyJean-Marie De Koninck and Imre KátaiAbstract. — In a series of papers, we constructed large families of normal numbers usingthe concatenation of the values of the largest prime factor P (n), as n runs through particularsequences of positive integers. A similar approach using the smallest prime factor function alsoallowed for the construction of normal numbers. Letting ω(n) stand for the number of distinctprime factors of the positive integer n, we then showed that the concatenation of the successivevalues of |ω(n)− blog log nc| in a fixed base q ≥ 2, as n runs through the integers n ≥ 3, yieldsa normal number. Here we prove the following. Let q ≥ 2 be a fixed integer. Given an integern ≥ n0 = max(q, 3), let N be the unique positive integer satisfying qN ≤ n < qN+1 and leth(n, q) stand for the residue modulo q of the number of distinct prime factors of n located inthe interval [log N, N ]. Setting xN := eN , we then create a normal number in base q using theconcatenation of the numbers h(n, q), as n runs through the integers ≥ xn0 .Résumé. — Dans une série d’articles, nous avons construit de grandes familles de nombresnormaux en utilisant la concaténation des valeurs successives du plus grand facteur premierP (n), où n parcourt certaines suites d’entiers positifs. Une approche similaire en utilisant lafonction plus petit facteur premier nous a aussi permis de construire d’autres familles de nombresnormaux. En désignant par ω(n) le nombre de nombres premiers distincts de n, nous avonsmontré que la concaténation des valeurs successives de |ω(n) − blog log nc| dans une base fixeq ≥ 2, où n parcourt les entiers n ≥ 3, donne place à un nombre normal. Ici, nous démontronsle résultat suivant. Soit q ≥ 2 un entier fixe. Étant donné un entier n ≥ n0 = max(q, 3), soitN l’unique entier positif satisfaisant qN ≤ n < qN+1 et désignons par h(n, q) le résidu moduloq du nombre de facteurs premiers distincts de n situés dans l’intervalle [log N, N ]. En posantxN := eN , nous créons alors un nombre normal dans la base q en utilisant la concaténation desnombres h(n, q), où n parcourt les entiers ≥ xn0 .Mathematical subject classification (2010). — 11K16, 11N37, 11N41.Key words and phrases. — Normal numbers, number of prime factors.Acknowledgements. — The research of the first author was supported in part by a grant from NSERC. Theresearch of the second author was supported by the Hungarian and Vietnamese TET 10-1-2011-0645.6 The number of large prime factors of integers and normal numbers1. IntroductionGiven an integer q ≥ 2, we say that an irrational number η is a q-normal number if theq-ary expansion of η is such that any preassigned sequence of length k ≥ 1, taken within thisexpansion, occurs with the expected limiting frequency, namely 1/qk.Even though constructing specific normal numbers is a very difficult problem, several authorspicked up this challenge. One of the first was Champernowne [2] who, in 1933, showed thatthe number made up of the concatenation of the natural numbers, namely the number0.123456789101112131415161718192021 . . . ,is normal in base 10. In 1946, Copeland and Erdős [4] proved that the same is true if onereplaces the sequence of natural numbers by the sequence of primes, namely for the number0.23571113171923293137 . . .In the same paper, they conjectured that if f(x) is any nonconstant polynomial whose valuesat x = 1, 2, 3, . . . are positive integers, then the decimal 0.f(1)f(2)f(3) . . ., where f(n) iswritten in base 10, is a normal number. Six years later, Davenport and Erdős [5] proved thisconjecture.Since then, many others have constructed various families of normal numbers. To name onlya few, let us mention Nakai and Shiokawa [15], Madritsch, Thuswaldner and Tichy [14] andfinally Vandehey [17]. More examples of normal numbers as well as numerous references canbe found in the recent book of Bugeaud [1].In a series of papers, we also constructed large families of normal numbers using the distribu-tion of the values of P (n), the largest prime factor function (see [6], [7], [8] and [9]). Recentlyin [10], we showed how the concatenation of the successive values of the smallest prime factorp(n), as n runs through the positive integers, can also yield a normal number. Letting ω(n)stand for the number of distinct prime factors of the positive integer n, we then showed thatthe concatenation of the successive values of |ω(n) − blog lognc| in a fixed base q ≥ 2, as nruns through the integers n ≥ 3, yields a normal number.Given an integer N ≥ 1, for each integer n ∈ JN := (eN , eN+1), let qN (n) be the smallestprime factor of n which is larger than N ; if no such prime factor exists, set qN (n) = 1. Fix aninteger Q ≥ 3 and consider the function f(n) = fQ(n) defined by f(n) = ` if n ≡ ` (mod Q)with (`,Q) = 1 and by f(n) = ε otherwise, where ε stands for the empty word. Then considerthe sequence (κ(n))n≥3 = (κQ(n))n≥3 defined by κ(n) = f(qN (n)) if n ∈ JN with qN (n) > 1and by κ(n) = ε if n ∈ JN with qN (n) = 1. Then, given an integer N ≥ 1 and writingJN = {j1, j2, j3, . . .}, consider the concatenation of the numbers κ(j1), κ(j2), κ(j3), . . ., thatis defineθN := Concat(κ(n) : n ∈ JN ) = 0.κ(j1)κ(j2)κ(j3) . . . .Then, set αQ := Concat(θN : N = 1, 2, 3, . . .) and let BQ = {`1, `2, . . . , `ϕ(Q)} be the set ofreduced residues modulo Q, where ϕ stands for the Euler function. In [11], we showed thatαQ is a normal sequence over BQ, that is, the real number 0.αQ is a normal number over BQ.Here we prove the following. Let q ≥ 2 be a fixed integer. Given an integer n ≥ n0 = max(q, 3),let N be the unique positive integer satisfying qN ≤ n < qN+1 and let h(n, q) stand for theresidue modulo q of the number of distinct prime factors of n located in the interval [logN,N ].Setting xN := eN , we then create a normal number in base q using the concatenation of thenumbers h(n, q), as n runs through the integers ≥ xn0 .Publications mathématiques de Besançon – 2015J.-M. De Koninck and I. Kátai 72. The main resultTheorem 2.1. — Let q ≥ 2 be a fixed integer. Given an integer n ≥ n0 = max(q, 3), let Nbe the unique positive integer satisfying qN ≤ n < qN+1 and let h(n, q) stand for the residuemodulo q of the number of distinct prime factors of n located in the interval [logN,N ]. Foreach integer N ≥ 1, set xN := eN . Then, Concat(h(n, q) : xn0 ≤ n ∈ N) is a q-ary normalsequence.Proof. — For each integer N ≥ 1, let JN = (xN , xN+1). Further let SN stand for the set ofprimes located in the interval [logN,N ] and TN for the product of the primes in SN . Letn0 = max(q, 3). Given a large integer N , consider the function(1) f(n) = fN (n) =∑p|nlogN≤p≤N1.Let us further introduce the following sequences:UN = Concat (h(n, q) : n ∈ JN ) ,V∞ = Concat (UN : N ≥ n0) = Concat (h(n, q) : n ≥ xn0) ,Vx = Concat (h(n, q) : xn0 ≤ n ≤ x) .Let us set Aq := {0, 1, . . . , q− 1}. If we fix an arbitrary integer r, it is sufficient to prove thatgiven any particular word w ∈ Arq, the number of occurrences Fw(Vx) of w in Vx satisfies(2) Fw(Vx) = (1 + o(1))xqr(x→∞).For each integer r ≥ 1, considering the polynomialQr(u) = u(u+ 1) · · · (u+ r − 1).and lettingρr(d) = #{u mod d : Qr(u) ≡ 0 mod d},it is clear that, since N is large,(3) ρr(p) = r if p ∈ SN .Observe that it follows from the Turán-Kubilius Inequality (see for instance Theorem 7.1 inthe book of De Koninck and Luca [12]), that for some positive constantC,(4)∑n∈JN(f(n)− log logN)2 ≤ CeN log logN.Letting εN = 1/ log log logN , it follows from (4) that(5) 1xN#{n ∈ JN : |f(n)− log logN | >1εN√log logN} → 0 (N →∞).This means that in the estimation of Fw(Vx), we may ignore those integers n appearing inthe concatenation h(2, q)h(3, q) . . . h(bxc, q) for which the corresponding f(n) is “far” fromlog logN in the sense described in (5).Let X be a large number. Then there exists a large integer N such that Xe< xN ≤ X. LettingL =]Xe,X], we writePublications mathématiques de Besançon – 20158 The number of large prime factors of integers and normal numbersL =]Xe, xN]∪ ]xN , X] = L1 ∪L2,say, and λ(Li) for the length of the interval Li for i = 1, 2.Given an arbitrary function δN which tends to 0 arbitrarily slowly, it is sufficient to considerthose L1 and L2 such that(6) λ(L1) ≥ δNX and λ(L2) ≥ δNX.The reason for this is that those n ∈ L1 (resp. n ∈ L2) for which λ(L1) < δNX (resp.λ(L2) < δNX) are o(x) in number and can therefore be ignored in the proof of (2).Let us first consider the set L2. We start by observing that any subword taken in the con-catenation h(n, q)h(n + 1, q) . . . h(n + r − 1, q) is made of co-prime divisors of TN (since notwo members of the sequence h(n, q), h(n+ 1, q), . . . , h(n+ r − 1, q) of r elements may havea common prime divisor p > logN). So, let d0, d1, . . . , dr−1 be co-prime divisors of TN andlet BN (L2; d0, d1, . . . , dr−1) stand for the number of those n ∈ L2 for which dj | n + j forj = 0, 1, . . . , r− 1 and such that(Qr(n),TNd0d1 · · · dr−1)= 1. We can assume that each of thedj ’s is squarefree, since the number of those n+j ≤ X for which p2 | n+j for some p > logNis  X∑p>logN1p2= o(X).In light of (4), we may assume that(7) ω(dj) ≤ 2 log logN for j = 0, 1, . . . , r − 1.By using the Eratosthenian sieve (see for instance the book of De Koninck and Luca [12])and recalling that condition (6) ensures that X − xN is large, we obtain that, as N →∞,BN (L2; d0, d1, . . . , dr−1) =X − xNd0d1 · · · dr−1∏p|TN/(d0d1···dr−1)(1− rp)+o xNd0d1 · · · dr−1∏p|TN/(d0d1···dr−1)(1− rp) .(8)Letting θN :=∏p|TN(1− rp), one can easily see that(9) θN = (1 + o(1))(log logN)r(logN)r (N →∞).Let us also introduce the strongly multiplicative function κ defined on primes p by κ(p) = p−r.Then, (8) can be written as(10) BN (L2; d0, d1, . . . , dr−1) =X − xNκ(d0)κ(d1) · · ·κ(dr−1)θN + o(xNκ(d0)κ(d1) · · ·κ(dr−1)θN)as N →∞. For each integer N > ee, letRN :=[log logN −√log logNεN, log logN +√log logNεN].Publications mathématiques de Besançon – 2015J.-M. De Koninck and I. Kátai 9Let `0, `1, . . . , `r−1 be an arbitrary collection of non negative integers < q. Note that thereare qr such collections. Our goal is to count how many times, amongst the integers n ∈ L2,we have f(n+ j) ≡ `j (mod q) for j = 0, 1, . . . , r−1. In light of (5), we only need to considerthose n ∈ L2 for whichf(n+ j) ∈ RN (j = 0, 1, . . . , r − 1).Let(11) S (`0, `1, . . . , `r−1) :=∑∗f(dj)≡`j mod qdj |TNj=0,1,...,r−11κ(d0)κ(d1) · · ·κ(dr−1),where the star over the sum indicates that the summation runs only on those dj satisfyingf(dj) ∈ RN for j = 0, 1, . . . , r − 1.From (10), we therefore obtain that#{n ∈ L2 : f(n+ j) ≡ `j mod q, j = 0, 1, . . . , r − 1}= (X − xN )θNS (`0, `1, . . . , `r−1) + o (xNθNS (`0, `1, . . . , `r−1))(12)as N →∞. Let us now introduce the functionη = ηN =∑p|TN1κ(p) .Observe that, as N →∞,η =∑logN≤p≤N1p(1− r/p) =∑logN≤p≤N1p+O ∑logN≤p≤N1p2= log logN − log log logN + o(1) +O( 1logN)= log logN − log log logN + o(1).(13)From the definition (11), one easily sees that(14) S (`0, `1, . . . , `r−1) = (1 + o(1))∑tj≡`j mod qtj∈RNηt0+t1+···+tr−1t0!t1! · · · tr−1!(N →∞),where we ignore in the denominator of the summands the factors κ(p)a (with a ≥ 2) sincetheir contribution is negligible.Moreover, for t ∈ RN , one can easily establish thatηt+1(t+ 1)! = (1 + o(1))ηtt! (N →∞)and consequently that, for each j ∈ {0, 1, . . . , r − 1},(15)∑tj≡`j mod qtj∈RNηtjtj != (1 + o(1))1q∑t∈RNηtt! = (1 + o(1))eηq(N →∞).Publications mathématiques de Besançon – 201510 The number of large prime factors of integers and normal numbersUsing (15) in (14), we obtain that(16) S (`0, `1, . . . , `r−1) = (1 + o(1))eηrqr(N →∞).Combining (12) and (16), we obtain that#{n ∈ L2 : f(n+ j) ≡ `j mod q, j = 0, 1, . . . , r − 1}= (X − xN )θNeηrqr+ o(xNθNeηrqr)= X − xNqr+ o(xN1qr)(N →∞),(17)where we used (9) and (13).Since the first term on the right hand side of (17) does not depend on the particular collection`0, `1, . . . , `r−1, we may conclude that the frequency of those integers n ∈ L2 for whichf(n + j) ≡ `j (mod q) for j = 0, 1, . . . , r − 1 is the same independently of the choice of`0, `1, . . . , `r−1.The case of those n ∈ L1 can be handled in a similar way.We have thus shown that the number of occurrences of any word w ∈ Arq in h(n, q)h(n +1, q) . . . h(n+ r − 1, q) as n runs over the bX −X/ec elements of L is (1 + o(1))(X −X/e)qr.Repeating this for each of the intervals]Xej+1,Xej](j = 0, 1, . . . , blog xc),we obtain that the number of occurrences of w for n ≤ x is (1 + o(1)) xqr, as claimed.The proof of (2) is thus complete and the Theorem is proved.3. Final remarksFirst of all, let us first mention that our main result can most likely be generalized in orderthat the following statement will be true:Let a(n) and b(n) be two monotonically increasing sequences of n for n = 1, 2, . . . suchthat n/b(n), b(n)/a(n) and a(n) all tend to infinity monotonically as n → ∞. Letf(n) stand for the number of prime divisors of n located in the interval [a(n), b(n)]and let h(n, q) be the residue of f(n) modulo q; then, the sequence h(n, q), n = 1, 2, ...,is a q-ary normal sequence.Secondly, let us first recall that it was proven by Pillai [16] (with a more general result byDelange [13]) that the values of ω(n) are equally distributed over the residue classes modulo qfor every integer q ≥ 2, and that the same holds for the function Ω(n), where Ω(n) :=∑pα‖n α.We believe that each of the sequences Concat(ω(n) (mod q) : n ∈ N) and Concat(Ω(n)(mod q) : n ∈ N) represents a normal sequence for each base q = 2, 3, . . .. However, the proofPublications mathématiques de Besançon – 2015J.-M. De Koninck and I. Kátai 11of these statements could be very difficult to obtain. Indeed, in the particular case q = 2,such a result would imply the famous Chowla conjecturelimx→∞1x∑n≤xλ(n)λ(n+ a1) · · ·λ(n+ ak) = 0,where λ(n) := (−1)Ω(n) is known as the Liouville function and where a1, a2, . . . , ak are kdistinct positive integers (see Chowla [3]).Thirdly, we had previously conjectured that, given any integer q ≥ 2 and letting resq(n) standfor the residue of n modulo q, it may not be possible to create an infinite sequence of positiveintegers n1 < n2 < · · · such that0.Concat(resq(nj) : j = 1, 2, . . .)is a q-normal number. However, we now have succeeded in creating such a monotonic sequence.It goes as follows. Let us define the sequence (mk)k≥1 bymk = f(k) + k!,where f is the function defined in (1). In this case, we obtain thatmk+1 −mk = k! · k + f(k + 1)− f(k),a quantity which is positive for all integers k ≥ 1 provided(18) f(k + 1)− f(k) > −k! · k,that is if(19) f(k) < k! · k.But since we trivially havef(k) ≤ ω(k) ≤ 2 log k ≤ k! · k,then (19) follows and therefore (18) as well.Hence, in light of Theorem 2.1, if we choose nk = mk, our conjecture is disproved.4. AcknowledgmentThe authors are grateful to Professor Andrzej Schinzel, since the problem examined hereoriginated from an email exchange between the second author and Professor Schinzel. Theauthors are also grateful to the referee whose comments helped improve the quality of thispaper.References[1] Y. Bugeaud, Distribution modulo one and Diophantine approximation. Cambridge Tracts inMathematics, 193. Cambridge University Press, Cambridge, 2012. xvi+300 pp.[2] D. G. Champernowne, The construction of decimals normal in the scale of ten, J. London Math.Soc. 8 (1933), 254-260.[3] S. Chowla, The Riemann Hypothesis and Hilbert’s Tenth Problem, Gordon and Breach, New York,1965.Publications mathématiques de Besançon – 201512 The number of large prime factors of integers and normal numbers[4] A. H. Copeland and P. Erdős, Note on normal numbers, Bull. Amer. Math. Soc. 52 (1946),857-860.[5] H. Davenport and P. Erdős, Note on normal decimals, Can. J. Math. 4 (1952), 58-63.[6] J. M. De Koninck and I. Kátai, On a problem on normal numbers raised by Igor Shparlinski,Bulletin of the Australian Mathematical Society 84 (2011), 337–349.[7] J. M. De Koninck and I. Kátai, Some new methods for constructing normal numbers, Annalesdes Sciences Mathématiques du Québec 36 (2012), no.2, 349–359.[8] J. M. De Koninck and I. Kátai, Construction of normal numbers using the distribution of thek-th largest prime factor, Bull. Australian Mathematical Society 88 (2013), 158-168.[9] J. M. De Koninck and I. Kátai, Using large prime divisors to construct normal numbers, AnnalesUniv. Sci. Budapest, Sect. Comput. 39 (2013), 45-62.[10] J. M. De Koninck and I. Kátai, Normal numbers generated using the smallest prime factor func-tion, Annales mathématiques du Québec (to appear).[11] J. M. De Koninck and I. Kátai, Constructing normal numbers using residues of selective primefactors of integers, Ann. Univ. Sci. Budapest, Sect. Comp., Volume 42, 2014 (to appear).[12] J. M. De Koninck and F. Luca, Analytic Number Theory: Exploring the Anatomy of Integers,Graduate Studies in Mathematics, Vol. 134, American Mathematical Society, Providence, RhodeIsland, 2012.[13] H. Delange, Sur la distribution des valeurs de certaines fonctions arithmétiques, Colloque surla Théorie des Nombres, Bruxelles, 1955, pp. 147–161. Georges Thone, Liège; Masson and Cie,Paris, 1956.[14] M. G. Madritsch, J. M. Thuswaldner and R. F. Tichy, Normality of numbers generated by thevalues of entire functions, J. of Number Theory 128 (2008), 1127-1145.[15] Y. Nakai and I. Shiokawa, Normality of numbers generated by the values of polynomials at primes,Acta Arith. 81 (1997), no. 4, 345-356.[16] S. S. Pillai, Generalisation of a theorem of Mangoldt, Proc. Indian Acad. Sci., Sect. A. 11 (1940),13–20.[17] J. Vandehey, The normality of digits in almost constant additive functions, Monatshefte für Math-ematik, 2012, 1-17.April 3, 2014Jean-Marie De Koninck, Département de mathématiques et de statistique, Université Laval,1045 Avenue de la médecine, Québec, Québec G1V 0A6, Canada • E-mail : jmdk@mat.ulaval.caUrl : www.jeanmariedekoninck.mat.ulaval.caImre Kátai, Computer Algebra Department, Eötvös Loránd University, 1117 Budapest, Pázmány PéterSétány I/C, Hungary • E-mail : katai@compalg.inf.elte.huPublications mathématiques de Besançon – 2015

The number of large prime factors of integers and normal numbers

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