USING LARGE PRIME DIVISORS TO CONSTRUCT NORMAL NUMBERS

Transcription

1 Annales Univ. Sci. Budaest., Sect. Com ) USING LARGE PRIME DIVISORS TO CONSTRUCT NORMAL NUMBERS Jean-Marie De Koninck 1 Québec, Canada) Imre Kátai 2 Budaest, Hungary) Dedicated to Professor Karl-Heinz Indlekofer on his seventieth anniversary Communicated by Ferenc Schi Received Setember 24, 2012; acceted October 10, 2012) Abstract. Given an integer q 2, a q-normal number is an irrational number ξ such that any reassigned sequence of l digits occurs in the q- ary eansion of ξ at the eected frequency, namely 1/q l. Let η) be a log η) slowly increasing function such that 0 as. Then, letting log P n) stand for the largest rime factor of n, set Qn) to be the smallest rime divisor of n which is larger than ηn), while setting Qn) = 1 if P n) > ηn). Then, we show that the real number 0.Q1)Q2)... is a normal number in base 10. With various similar constructions, we create large families of normal numbers in any given base q 2. Finally, we consider eonential sums involving the Qn) function. 1. Introduction Given an integer q 2, a q-normal number, or simly a normal number, is an irrational number whose q-ary eansion is such that any reassigned sequence, of length l 1, of base q digits from this eansion, occurs at the eected frequency, namely 1/q l. Key words and hrases: Normal numbers, largest rime factor, smallest rime factor Mathematics Subject Classification: 11K16, 11N37, 11A41. 1 Research suorted in art by a grant from NSERC. 2 Research suorted by ELTE IK and by the Hungarian and Vietnamese TET grant agreement no. TET ).

2 46 J.-M. De Koninck and I. Kátai Let A q := {0, 1,..., q 1}. Given an integer l 1, an eression of the form i 1 i 2... i l, where each i j A q is called a word of length l. We sometimes write λβ) = l to indicate that β is a word of length l. The symbol Λ will denote the emty word. We let A l q stand for the set of all words of length l and A q stand for the set of all the words regardless of their length. Given a ositive integer n, we write its q-ary eansion as 1.1) n = ε 0 n) + ε 1 n)q + + ε t n)q t, where ε i n) A q for 0 i t and ε t n) 0. associate the word 1.2) n = ε 0 n)ε 1 n)... ε t n) = ε 0 ε 1... ε t A t+1 q. To this reresentation, we Let P n) stand for the largest rime factor of n 2, with P 1) = 1. In a recent aer [5], we showed that if F Z[] is a olynomial of ositive degree with F ) > 0 for > 0, then the real numbers and 0.F P 2)) F P 3))... F P n))... 0.F P 2 + 1)) F P 3 + 1))... F P + 1))..., where runs through the sequence of rimes, are q-normal numbers. Let η) be a slowly increasing function, that is an increasing function ηc) satisfying lim = 1 for any fied constant c > 0. Being slowly increasing, η) log η) it satisfies in articular the condition 0 as. log We then let Qn) be the smallest rime divisor of n which is larger than ηn), while setting Qn) = 1 if P n) > ηn). Then, we show that the real number 0.Q1) Q2) Q3)... is a q-normal number. With various similar constructions, we create large families of normal numbers in any given base q 2. Finally, we consider eonential sums involving the Qn) function. 2. Main results Theorem 1. Given an arbitrary basis q 2 and for any integer n, let n be as in 1.2). Then the number is a q-normal number. ξ 1 = 0.Q1) Q2) Q3)...

3 Using large rime divisors to construct normal numbers 47 Let stand for the set of all rimes. Given an integer q 2, let R, 0, 1,..., q 1 be disjoint sets of rime numbers such that = R 0 1 q 1, and such that, uniformly for 2 v u as u, π[u, u + v] j ) = 1 ) u q π[u, u + v]) + O log 5 u so that, in articular, ) u π[u, u + v] R) = O log 5. u Then, consider the function κ defined on as follows: { l if l, κ) = Λ if R. With this notation, we have Theorem 2. The number is a q-normal number. ξ 2 = 0.κQ1))κQ2))κQ3))... j = 0, 1,..., q 1), Remark 1. In an earlier aer [4], we used such classification of rime numbers to create normal numbers, but by simly concatenating the numbers κ1), κ2), κ3),.... Let a be a fied non zero integer. Then we have the following result. Theorem 3. The number ξ 3 = 0.κQ2 + a))κq3 + a))κq5 + a))... κq + a))..., where runs through the set of rimes, is a q-normal number. Define as the set of all the rime numbers 1 mod 4). Then, let R, 0, 1,..., q 1 be disjoint sets of rime numbers such that = R 0 1 q 1, and such that, uniformly for 2 v u as u, π[u, u + v] j ) = 1 ) u q π[u, u + v] ) + O log 5 u j = 0, 1,..., q 1),

4 48 J.-M. De Koninck and I. Kátai so that, in articular, ) u π[u, u + v] R ) = O log 5. u Then, consider the following function defined on as follows { l if ν) = l, Λ if q 1 l=0 l. With this notation, we have the following result. Theorem 4. The number is a q-normal number. ξ 4 = 0.νQ1))νQ2))νQ3))... Consider the arithmetic function fn) = n Then, we have the following result. Theorem 5. The two numbers ξ 5 = 0.κQf1)))κQf2)))κQf3)))..., ξ 6 = 0.κQf2)))κQf3)))κQf5)))... κqf)))..., where runs through the set of rimes, are q-normal numbers. Remark 2. One can show that this last result remains true if fn) is relaced by another non constant irreducible olynomial. We now introduce the roduct function F n) = nn + 1) n + q 1). Observe that if for some ositive integer n, we have = QF n)) > q, then n+l only for one l {0, 1,..., q 1}, imlying that l is uniquely determined for all ositive integers n such that QF n)) > q. Thus we may define the function { l if = QF n)) > q and n + l, τn) = Λ otherwise. Using this notation, we have the following result. Theorem 6. The number is a q-normal number. ξ 7 = 0.τq + 1)τq + 2)τq + 3)...

5 Using large rime divisors to construct normal numbers 49 We now introduce the roduct function Gn) = n + 1)n + 2) n + q) and further define the function { l if = QGn)) > q + 1 and n + l + 1, ρn) = Λ otherwise. Moreover, let j ) j 1 be the sequence of all rimes larger than q, that is, q < 1 < 2 < With this notation, we have the following result. Theorem 7. The number is a q-normal number. ξ 8 = 0.ρ 1 )ρ 2 )ρ 3 )... Let α be an arbitrary irrational number. We will be using the standard notation ey) = e{2πiy}. We then have the following. Theorem 8. Let A) := n fn)eαqn)), where f is any given multilicative function satisfying fn) = 1 for all ositive integers n. Then, A) 2.1) lim = Notation and reliminary lemmas log n For each integer n 2, let Ln) =. Let β A l q and n be written log q as in 1.1). We then let ν β n) stand for the number of occurrences of the word β in the q-ary eansion of the ositive integer n, that is, the number of times that ε j n)... ε j+l 1 n) = β as j varies from 0 to t l 1). The letters and π will always denote rime numbers. The letter c with or without subscrit always denotes a ositive constant but not necessarily the same at each occurrence. We will be using a key result obtained by Bassily and Kátai [1] and which we state here as two lemmas, a roof of which, in a more general contet, can be found in our earlier aer [5].

6 50 J.-M. De Koninck and I. Kátai Lemma 1. Let κ u be a function of u such that κ u > 1 for all u. Given a word β A l q and setting { V β u) := # : u 2u such that then, there eists a ositive constant c such that cu V β u) log u)κ 2. u ν β) Lu) > κ u q l Lu) }, Lemma 2. Let κ u be as in Lemma 1. Given β 1, β 2 A l q with β 1 β 2, set } β1,β 2 u) := # { : u 2u such that ν β1 ) ν β2 ) > κ u Lu). Then, for some ositive constant c, cu β1,β 2 u) log u)κ 2. u 4. Proof of Theorem 2 We start by roving Theorem 2 since its content will be useful for the roof of Theorem 1. Let I = [, 2] and first observe that #{n I : there eists n, [η), η2)]} ) 2 c η) η2) = o1) ). η) η2) 1 = This means that with the ecetion of o) integers n I, the number Qn) is the smallest rime divisor of n bigger than η). Secondly, observe that we may assume that, given any fied small ε > 0, we may assume that Qn) η) 1/ε. Indeed, 4.1) #{n I : Qn) > η) 1/ε } 1 1 ) ε. η)< η) 1/ε

7 Using large rime divisors to construct normal numbers 51 Now let 0, 1,..., k 1 be any distinct rimes belonging to the interval η), η) 1/ε ), and let 0 < 1 < < k 1 be the unique ermutation of the rimes 0, 1,..., k 1, namely the one such that has all its members aear in increasing order, so that we have η) < 0 < 1 < < k 1 < η) 1/ε. Our first goal will be to estimate the size of N 0, 1,..., k 1 ) := #{n : Qn + j) = j, j = 0, 1,..., k 1}. We must therefore estimate the number of those integers n I for which j n + j j = 0, 1,..., k 1), while at the same time π j, n + j) = 1 if η) < < π j < j j = 0, 1,..., k 1). Before moving on, let us set Q k = 0 1 k 1 and T j = π j = 0, 1,..., k 1). η)<π< j It is then easy to see that, say by using the Eratosthenian sieve see for instance Chater 12 in the book of De Koninck and Luca [2]), we obtain 4.2) N 0, 1,..., k 1 ) = 1 + o1)) Q k Σ 0 ), where Σ 0 = δ 0,...,δ k 1 δ j T j j=0,1,...,k 1) δ i,δ j )=1 if i j µδ 0 ) µδ k 1 ) δ 0 δ k 1 here µ stands for the Möbius function). One can see that Σ 0 = = η)<π< 0 1 k ) π 0 <π< 1 1 k 1 ) π k 2 <π< k 1 4.3) ) log k ) = 1 + o1)) 0 log k+1 1 log ) 1 log η) log k 1 0 log. k 2 log η) Hence, if we set σ) :=, it follows from 4.3) that log 4.4) Σ 0 = 1 + o1))σ 0 ) σ k 1 ) ). Substituting 4.4) in 4.2), we obtain k 1 4.5) N 0, 1,..., k 1 ) = 1 + o1)) j=0 σ j ) j ), 1 1 ) π an estimate which holds uniformly for η) j η) 1/ε j = 0, 1,..., k 1).

8 52 J.-M. De Koninck and I. Kátai We will now use a technique which we first used in [3] to study the distribution of subsets of rimes in the rime factorization of integers. We first introduce the sequence u 0 = η), u j+1 = u j + u j log 2 u j for each j = 0, 1, 2,... and then let T be the unique ositive integer satisfying u T 1 < η) 1/ε u T. Then, consider the intervals J 0 := [u 0, u 1 ), J 1 := [u 1, u 2 ),..., J T 1 := [u T 1, u T ). Choose k arbitrary integers j 0,..., j k 1 {0, 1,..., T 1}, as well as k arbitrary integers i 0,..., i k 1 from the set {0, 1,..., q 1}, and consider the quantity 4.6) M j ) 0, j 1,..., j k 1 = N i 0, i 1,..., i 0,..., k 1 ). k 1 l J l il Observe that σ h) = 1 + o1)) σu h) as if J h. It follows from h u h this observation and using 4.5) and 4.6) that 4.7) M j 0, j 1,..., j k 1 i 0, i 1,..., i k 1 ) = 1 + o1)) k 1 l J l il j=0 σu j ) u j. Using Theorem 1 of our 1995 aer [3] in combination with 4.7), we obtain that M 4.8) j ) 0, j 1,..., j k 1 = 1 + o1))m i 0, i 1,..., i k 1 j ) 0, j 1,..., j k 1 i 0, i 1,..., i k 1 ), where i 0, i 1,..., i k ) is any arbitrary sequence of length k comosed of integers from the set {0,..., q 1}. Finally, consider the eression A := κq ))... κq 2 1)). It follows from 4.8) that, for any given word β A k q, the number of occurrences of β as a subword in the word A is equal to 1 + o1)) q k comleting the roof of Theorem 2. as, thus

9 Using large rime divisors to construct normal numbers Proof of Theorem 1 Let B = Q )... Q 2 1). Also, let Q n) = min and observe that Q n) Qn), while if Q n) n >η) Qn), then n if η) < < η2). Moreover, let B = Q )... Q 2 1). Clearly, we have, since η) was chosen to be a slowly oscillating function, 5.1) 0 λb ) λb ) c η)<<η2) ). log log q c 1 log η2) η) = o) It follows from 5.1) that we now only need to estimate λb ). To do so, we first let δ be a function tending to 0 very slowly as, in a manner secified below. If < δ, we have 5.2) R ) : = #{n I : Q n) = } = = 1 + o1)) 1 1 ) = π = 1 + o1)) η)<π< log η) log while on the other hand, if δ 2, we have 5.3) R ) < c Now, observe that, as, λb ) = η)< 2 5.4) = 1 + o1)) log q R )λ) = η)< 2 log η) = 1 + o1)) log log q η)< 2 log η) log η) log. R ) log log η) + O ), log = log q 1 + O log η) δ < log η) log 1 ). δ

10 54 J.-M. De Koninck and I. Kátai Choosing the function δ in such a way that log allows us to relace 5.4) with log 1 δ = o log log η) ) 5.5) λb) log η) = 1 + o1)) log log log q log η) ). Now, let β 1, β 2 A k q. We will now make use of Lemmas 1 and 2. For this, we first write T [η), δ ] = I uj, where I uj = [u j, u j+1 ), with u 0 = η), u j = 2 j η) for j = 1, 2,..., T +1, where T is defined as the unique ositive integer satisfying u T < δ u T +1. if In the sirit of Lemma 1, we will say that the rime I u is a bad rime ma ν β) Lu) > κ u Lu) and a good rime if β A l q q l j=0 ν β) Lu) q l κ u Lu). We will now searate the sum R )λ) running over the rimes located in the intervals [u j, u j+1 ) into two categories, namely the bad rimes and the good rimes. 5.6) First, using 5.2) and 5.3), we have [u j,u j+1 ) bad R )λ) cκu j ) [u j,u j+1) log η) log η) + j log 2. log η) log On the other hand, if is a good rime, one can easily establish that the number of occurrences of the words β 1 and β 2 in the word B are close to each other, in the sense that 5.7) ν β1 B ) ν β2 B ) = oλb )).

11 Using large rime divisors to construct normal numbers 55 Hence, roceeding as in [5], it follows, considering the true size of λb) given by 5.5) and in light of 5.1), 5.6) and 5.7), that the number of words β A k q aearing in B is equal to 1 + o1)) λb ) q k as. We then roceed in a same manner to obtain similar estimates successively for the intervals I /2, I /2 2,... Thus, reeating the argument used in [5], Theorem 1 follows immediately. The roofs of Theorems 3 through 7 can be obtained along the same lines and will therefore be omitted. 6. Proof of Theorem 8 6.1) To rove this theorem, we will consider two cases searately. Let us first assume that R1 f) iτ ) < for some real number τ. It can be roved as we did in [6]) that one can assume that τ = 0. For a start, define the additive function u imlicitly on rime owers by f β ) = e iuβ). Then, for each large number D, define the multilicative function f D on rime owers by { f D β f ) = β ) if D, 1 if > D. In light of 6.1), we have that 6.2) u 2 ) <. Further set a D ) = D< u) 1, b2 D) = D< u 2 ). Since fn) = f D n) e i u β ) = f Dn) e {iu D n)}, β n

12 56 J.-M. De Koninck and I. Kátai say, then, by using the Turán Kubilius inequality, we obtain that A) A D ) = O b D )), where A D ) = η D ) n f D n)eαqn)), where η D ) = e ia D). Further define the function τ D imlicitly by the equation f D n) = d n τ Dd). It is clear that τ D d) = 0 if d, D) > 1, while τ D β ) 2 for all rime owers β. We clearly have 6.3) A D ) = η D ) say. On the other hand, 1 P d) D P d) D τ D d) md τ D d) Σ d Therefore, for some k D, we have where ρ D 0 as D. 1 P d) D Let us now consider the sum 6.4) T Y = eαqmd)) = η D ) τ D d) d d>k D τ D d) Σ d ρ D, Y m 2Y eαqm)). D P d) D ). 1 τ D d)σ d, Recall that Qm) is the smallest rime divisor of m which is larger than ηm). Now, consider the somewhat similar function Q 1 m), which stands for the smallest rime divisor of m which is larger than η). Recalling the argument used at the beginning of the roof of Theorem 2, we easily see that 6.5) #{m [Y, 2Y ] : Q 1 m) Qm)} = cy log η2y ) ηy ) = oy ) as Y. Therefore, setting T 1) Y = Y m 2Y eαq 1 m)),

13 Using large rime divisors to construct normal numbers 57 it is clear that TY T 1) Y = oy ) Y ). Moreover, as Y, we have 6.6) #{m [Y, 2Y ] : Q 1 m) = } = 1 + o1)) Y = 1 + o1)) Y Similarly as we obtained 4.1), we easily rove that ηy )<π< log ηy ) log. 6.7) #{m [Y, 2Y ] : Qm) > ηy ) 1/ε } εy. On the other hand, using 6.4), 6.6) and 4.1), we have eα) log ηy ) 6.8) T Y = Y + OεY ). log ηy )<<ηy ) 1/ε 1 1 ) = π By using the well known I.M. Vinogradov theorem [10] asserting that lim 1 π) eα) = 0, we obtain from 6.8) that 6.9) T Y Y ε + o1) Y ). Using this, we can estimate Σ d. Indeed, we have 6.10) Σ d eαqdm)) + 2 L d. 2 L d <m< d Let l D be an arbitrary large number and choose L so that ) η 2 L > l D. d Note that for an arbitrary large L, this inequality will hold rovided is large enough. Alying 6.9), it follows from 6.10) that 6.11) Σ d 2 L d + cε d.

14 58 J.-M. De Koninck and I. Kátai Using 6.11) in 6.3), we obtain that 6.12) A D ) cε + 1 ) 2 L ) + τ D d). 1 d D d>l D Since D and L were chosen to be arbitrary numbers, it follows from 6.12) that A D ) 6.13) lim = 0. Since A) = A D) + O b D )) and recalling the definition of b D ) and estimate 6.2), it follows from 6.13) that A) lim su cb D ) = o1), so that if D, we immediately obtain 2.1) for the first case, that is when 6.1) holds. 6.14) It remains to consider the case R1 f) iτ ) First, it is clear that, using 6.5), we have 6.15) say. E) : = <n 2 = <n 2 = <n 2 = E 1 ) + o), = for all real numbers τ. fn)eαqn)) = fn)eαq 1 n)) + <n 2 Q 1 n) Qn) fn)eαq 1 n)) + o) = fn)eαqn)) = In light of 6.7), we may ignore those n, 2] for which Q 1 n) > η) 1/ε, that is, 6.16) fn)eαq 1 n)) ε. <n 2 Q 1 n)>η) 1/ε

15 Using large rime divisors to construct normal numbers 59 Combining 6.15) and 6.16), we can write that 6.17) E) = where, setting Π := η)<<η) 1/ε f)eα)σ + Oε), η)<π< 6.18) Σ = π, <m 2 m,π)=1 fm). Now, consider the summation S) = n fn). In light of 6.14), it follows from a classical theorem of Halász see [9]) that there eists a function ε) which tends to 0 monotonically as such that S) ε), which in turn imlies that 6.19) S2) S) ε). From 6.18), we get that 6.20) Σ = <m 2 = δ Π µδ) fm) = δ Π µδ)fδ) δ Π,m) <mδ 2 S µδ) = fmδ) = 2 δ ) )) S + Er, δ where Er is the error term coming from those terms for which m, δ) > 1. Thus, it follows from 6.19) and 6.20) that 6.21) Σ ) µ 2 δ)ε + Er µ2 δ) + Er, δ δ δ Π δ Π

16 60 J.-M. De Koninck and I. Kátai where we used the fact that since ma η)<<η) 1/ε δ Π δ 0 as, then ε/δ) = o/δ) uniformly for η) < < η) 1/ε and δ Π. Now, since µ2 δ) δ δ Π η)<π<ηn) 1/ε it follows from 6.21) that, as, 6.22) Σ c ε o1) + Er ) c 1 π ε, Using 6.22) in 6.17), we obtain that, as, 6.23) E) c ε 1 o1) + V ) + Oε), η)<<η) 1/ε where We will now show that V ) = η)<<η) 1/ε Er. 6.24) V ) = o) ). Setting J = J) = η), η) 1/ε ) and writing those mδ such that m, δ) > 1 as mδ = lκ 2 δ 1, where κ and δ 1 are squarefree numbers whose rime factors all belong to J, we have that 6.25) V ) κ η) = κ η) c κ η) µ 2 κ) µ 2 κ) <lκ 2 δ 1 2 J π δ 1 π J J π δ 1 π J µ 2 κ) κ 2 J π δ 1 π J µ 2 δ 1 ) = µ 2 δ 1 ) κ 2 <l 2 δ 1 κ 2 δ 1 µ 2 δ 1 ) δ 1. 1

17 Using large rime divisors to construct normal numbers 61 Since it is easily checked that π δ 1 π J J 1 c 1 log 1 ε, µ 2 δ 1 ) δ 1 π J κ η) 1 κ 2 c 3 η), then using these estimates in 6.25), we obtain that ) c 2 log η), π ε V ) c 4 log η) 1 η) ε log 1 ε = o) ), thus roving our claim 6.24). Substituting 6.24) in 6.23), we obtain that E) c 1 ε log 1 o1) + o) + Oε) = o) ε ), from which it follows that given any arbitrarily small number ξ > 0, there is some 0 = 0 ξ) such that 6.26) EX) ξ X for all X > 0. Therefore, given any fied large number and letting L be the smallest integer such that 2 L > /2, we have that, using 6.26) reetitively, L ) L A) = E 2 a cξ 2 a < cξ, a=1 a=1 thus roving 2.1) in the second case, as requested. This comletes the roof of Theorem 8. References [1] Bassily, N.L. and I. Kátai, Distribution of consecutive digits in the q-ary eansions of some sequences of integers, Journal of Mathematical Sciences, 781) 1996),

18 62 J.-M. De Koninck and I. Kátai [2] De Koninck, J.M. and F. Luca, Analytic Number Theory: Eloring the Anatomy of Integers, Graduate Studies in Mathematics, Vol. 134, American Mathematical Society, Providence, Rhode Island, [3] De Koninck, J.M. and I. Kátai, On the distribution of subsets of rimes in the rime factorization of integers, Acta Arithmetica, 722) 1995), [4] De Koninck, J.M. and I. Kátai, Construction of normal numbers by classified rime divisors of integers, Functiones et Aroimatio, 452) 2011), [5] De Koninck, J.M. and I. Kátai, On a roblem on normal numbers raised by Igor Sharlinski, Bulletin of the Austr. Math. Soc., ), [6] De Koninck, J.M. and I. Kátai, Eonential sums involving the largest rime factor function, Acta Arithmetica, ), [7] De Koninck, J.M. and I. Kátai, Construction of normal numbers using the distribution of the k-th largest rime factor, Bull. Australian Mathematical Society, to aear. [8] Halász, G., Über die Mittelwerte multilikativen zahlentheoretischer Funktionen, Acta Math. Acad. Scient. Hungar., ), [9] Halberstam, H.H. and H.E. Richert, Sieve Methods, Academic Press, London, [10] Vinogradov, I.M., The Method of Trigonometric Sums in the Theory of Numbers, translated, revised and annotated by A. Davenort and K.F. Roth, Interscience, New York, J.-M. De Koninck Déartment de mathématiques et de statistique Université Laval Québec Québec G1V 0A6 Canada jmdk@mat.ulaval.ca I. Kátai Deartment of Comuter Algebra Faculty of Informatics Eötvös Loránd University Pázmány Péter sétány 1/C H-1117 Budaest, Hungary katai@comalg.inf.elte.hu