On the Fractional Parts of a n /n

Transcription

1 On the Fractional Parts of a n /n Javier Cilleruelo Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM and Universidad Autónoma de Madrid Madrid, Spain franciscojavier.cilleruelo@uam.es Angel Kumchev Department of Mathematicss, Towson University Towson, MD 2252, U.S.A. akumchev@towson.edu Florian Luca Instituto de Matemáticas Universidad acional Autonoma de México C.P , Morelia, Michoacán, Mexico fluca@matmor.unam.mx Juanjo Rué Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM C/ icolás Cabrera 3-5, Madrid, Spain juanjo.rue@icmat.es Igor E. Shparlinski Deptartment of Computing, Macuarie University Sydney, SW 209, Australia igor.shparlinski@m.edu.au

2 Abstract We give various results about the distribution of the seuence {a n /n} n modulo, where a 2 is a fixed integer. In particular, we find and infinite subseuence A such that {a n /n} n A is well distributed. Also we show that for any constant c > 0 and a sufficiently large, the fractional parts of the first terms of this seuence hit every interval J [0, ] of length J c Introduction. Motivation Let a 2 be an integer. It has been asked [] whether for every nonzero integer h, we have the estimate e h an = o n n as, where, as usual, for a real number x we put ex = exp2πix. It is well-known that the bound is euivalent to the uniform distribution modulo of the seuence {a n /n} n, where {γ} denotes the fractional part of a real number γ. While we certainly believe that holds, we have not been able to confirm this conjecture..2 Our results Instead, here we settle for the somewhat more modest goal of showing that the seuence {a n /n} n is dense modulo. This statement is an immediate conseuence of a stronger result which we obtain here, that asserts that the seuence {a n /n} is uniformly distributed when we restrict n to a certain subset of the positive integers. We define the set A = {p : p, primes, log p/log a} and put A = A [, ]. We recall that the discrepancy Γ of a seuence Γ of M points Γ = { } γ m M m= 2 2

3 in the unit interval [0, ] is defined as Γ = sup T Γ [α, β] β α M, [α,β] [0,] where T Γ [α, β] is the number of points of Γ inside the interval [α, β]. Our first result provides bounds for the discrepancy of {a n /n} n A, showing that the seuence is dense in [0, ]: Theorem. The discrepancy D of the seuence {a n /n} n A is D = O log log log log. log log log Since it is also interesting to derive strong explicit bounds on the density, we address this issue too. Our second result estimates the length of intervals in which we can assure the existence of points of the seuence for sufficiently large: Theorem 2. For any constant c > 0 and a sufficiently large, every interval J [0, ] of length J c contains a number of the form {a n /n} for some n. If we assume some unproved hypothesis, say the Generalized Riemann Hypothesis or the Generalized Lindelöf Hypothesis, we can reduce the size of the intervals in Theorem 2 to J /2+ϵ for any fixed ϵ > 0. Furthermore, by combining the argument of Theorem 2 with a slight generalization of a result of K. Matomäki [7] to differences of primes in a progression with a small modulus, one can show that the total measure of gaps larger than /2 is O /3..3 otation We use the Landau symbol O and o as well as the Vinogradov s symbols, and. Recall that A = OB, A B and B A are all euivalent to the fact that the ineuality A cb holds with some constant c. Furthermore, A B means that both A B and B A hold. Throughout the paper, p and always denote prime numbers. For two integers u and v, their greatest common divisor is denoted by u, v. 3

4 As usual, for relatively prime integers a and we denote by ord a the multiplicative order of a in Z/Z. We use πx for the number of primes p x, and for coprime positive integers k and r we use πx; k, r for the number of primes smaller than or eual to x in the arithmetic progression r mod k. Finally, we denote by φn the Euler function and by P n the largest prime divisor of an integer n we set P =. 2 Preliminaries 2. Some general facts We use the asymptotic estimate that follows from the Siegel Walfisz Theorem, see [4, Corollary 5.29], πx; k, r = πx φk + O x 3 log x A valid for any k with k, r =, and any constant A > 0. We also need the bound given by the Brun-Titchmarsh Theorem, see [4, Theorem 6.6] πx; k, r x φk log x/k valid for all x > k. 4 We recall the Mertens Formula for the sum of reciprocals of the primes p x in the following crude form x We also use the following well known lower bound φn x = log log x + O. 5 n log log n for n 3. 6 One of our main tools is the classical Erdős-Turán Ineuality see, for example, [2, Theorem.2] that relates the uniformity of distribution to exponential sums. 4

5 Lemma. For any integer L, for the discrepancy Γ of the seuence 2, we have Γ L + M e hγ m M h. 0< h L To prove Theorem 2, we need the following slight modification of [3, Theorem 0.8]. Lemma 2. There exists an absolute constant ϑ < with the following property: For any A > 0, if x is sufficiently large, log x A, l, = and x ϑ h x, then πx;, l πx h;, l m= h 20ϕ log x. Proof. We recall that [3, Theorem 0.8] states such a result with ϑ = and the constant 0.09 in place of /20 in the lower bound. In fact, the proof of [3, Theorem 0.8] yields a lower bound of the form πx;, l πx h;, l Cθh ϕ log x, for x θ h x. Here, Cθ is a continuous decreasing function of θ such that C It is clear from the continuity of Cθ see also the comments in [3, 7.0] that one also has Cθ for some θ 0 < Taking ϑ = θ 0, one obtains the lemma. 2.2 Some facts about A Lemma 3. We have #A Proof. We observe that if p A then log log log. log a p /. 5

6 Let Q be the largest prime such that a / and notice that Q log /log a. We observe that Thus, we have #A = Q On the other hand, Q πa Q Q π/ Q and the result follows. a aq Q Q 2 log 2. 7 π/ πa = π/ + O log 2. Q log/ log Q log log log log For a pair of primes p > we define u p by the condition u pp mod, u p. 8 For real α and β, we also write α β mod if α β Z. Lemma 4. For primes p >, we have a p p ap a u p + a p mod. Proof. By 8, we have and then a p p a p u p mod, a p p = ap a p+ p ap+ p a ap u p which concludes the proof. + a a a p p + a p p ap a u p + a p mod, 6

7 3 Proofs of the Main Results 3. Proof of Theorem The core of the proof is based on estimating the exponential sum ha p Sh, = e p p A a p a = e h u p + a = p Sh, + E, where and p A Sh, = n=p A E e h ap a u p p A h a p. Let Q be defined as in the proof of Lemma 3. formula 5 to get E Q h a p p We use 7 and Mertens h log log log 2. 9 ow we fix a prime and a pair of integers u, v with u, v and u, = v, =. By the Chinese Remainder Theorem, we see that the primes p which satisfy up mod, p v mod, belong to a certain arithmetic progression z u, v mod. Thus, Sh, = Q 2 u= v= v, = π e h av a u ;, z u, v π a ;, z u, v. 7

8 Using 3 with A = 2 and 4 and noticing that the sum over u and v contains φ terms, we obtain Sh, = Σ 0 + OΣ + Σ 2, 0 where Σ 0 = Q π/ φ a u= 2 v= v, = e h av a u, log 2, 2 Σ = Q Σ 2 = log log log 2 log / log 2. 3 Q We continue getting estimates for Σ 0. Write sh, = u= 2 v= v, = e We observe that for with, ah = we have u= e h av a u = h av a u. { if a v mod, if a v mod. From this observation, for a prime such that, ah = we have the exact value sh, = φ + #{v : v, =, a v mod }. Thus, sh, φ + ord a 2 ord a. 4 8

9 We also have the trivial bound sh, φ if, ah >. Therefore, Σ 0 2 Q ah + Q ah log φ log/ ord a φ φ log/ Q φ ord a + log. Q ah Using the trivial bound ord a log / log a and the bound 6, we obtain Q φ ord a log log log = O. where in the last sum the summation is taken over all prime numbers. Hence, Σ log h Substituting the bound 5 together with the bound 2 and 3 in 0, taking into account 9 and recalling the bound for #A obtained in Lemma 3 we get We now take Sh, h log log log 2 L = + + log h log log log log log log log. and apply Lemma with M = #A to derive that the discrepancy 9

10 D of the seuence A satisfies the bound D L + M L + L M 0< h L which concludes the proof. log log log 2 Sh; h + log L M log + M log log log log log log log log + log log log log log log log log log log + log L log log log 3.2 Proof of Theorem 2 0< l L/ 0< h L l log log log log log log log, h h Let be large. We consider the largest prime log /log a and write T = a /. We fix some c > 0 and prove that any interval J of length J c contains some element of the form {a p /p} for some prime p in the set P = {p, p mod, T/2 p T }. Lemma 2 implies that consecutive primes in this set satisfy p n+ p n T ϑ, with some constant ϑ < We observe also that if p mod then a p mod. Thus a p mod p. Then we have that a p p ap a p a p T p mod. Finally, we observe that if p runs in P, then T/p [, 2] and that consecutive fractional parts, corresponding to consecutive primes p n, p n+ P satisfy { } { } T T T T 0 < = p n p n+ p n p n+ which concludes the proof. = T p n T p n+ = T p n+ p n p n p n+ T ϑ = o 0.475, 0

11 Acknowledgments This paper started while F. L. visited the Mathematical Institute of the UAM in Madrid in April of 20 and continued during a visit of F. L. at the Department of Computing, Macuarie University, Sydney, in ovember 20. He thanks the people of these institutions for their hospitality. During the preparation of this paper, J. C. and J. R. was supported by Grant MTM of MICI Spain, J. R. was also supported by JAE-DOC Grant of CSIC Spain and I. S. was supported in part by ARC Grant DP Australia and by RF Grant CRP Singapore References [] W. D. Banks, M. Z. Garaev, F. Luca and I. E. Shparlinski, Uniform distribution of fractional parts related to pseudoprimes, Canadian J. Math , [2] M. Drmota and R. Tichy, Seuences, discrepancies and applications, Springer-Verlag, Berlin, 997. [3] G. Harman, Prime-Detecting Sieves, LMS Monographs 33, Princeton University Press, [4] H. Iwaniec and E. Kowalski, Analytic number theory, Amer. Math. Soc., Providence, RI, [5] P. Kurlberg and C. Pomerance, On the period of the linear congruential and power generators, Acta Arith., , [6] F. Luca and I. E. Shparlinski, Quadratic fields generated by the Shanks seuence, Proc. Edinb. Math. Soc., , [7] K. Matomäki, Large differences between consecutive primes, Quart. J. Math. Oxford, ,