Products of ratios of consecutive integers

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1 Products of ratios of consecutive integers Régis de la Bretèche, Carl Pomerance & Gérald Tenenbaum 27/1/23, 9h26 For Jean-Louis Nicolas, on his sixtieth birthday 1. Introduction Let {ε n } 1 n<n be a finite sequence with each ε n {, ±1}, and write a b = n εn, n n<n where the fraction is in its smallest terms. Now, define AN as the maximal value of a as {ε n } 1 n<n runs through all possible 3 N 1 sequences of, ±1. One might also consider the maximal value of b, but this is the same. We obviously have AN N!, hence log AN N log N for all N. In [6], it is shown by an elegant near-tiling of the integers in [1, N] with triples n, 2n, 2n + 1 that log AN { o1} N log N. Further, a brief argument of M. Langevin is presented that log AN {log 4 + o1}n. Our aim in this article is to establish the true order of magnitude for log AN. Put kc := log1 2c 2 c log 1 + 2c2, 1 3c Kc := 2 Theorem 1.1. For large N, we have ku du, K := max Kc.175. <c<1/5 1 1 log AN {K + o1}n log N. Let P n denote the largest prime factor of a positive integer n with the convention that P 1 = 1. The lower bound 1 1 is an easy consequence of the estimate stated in the following result. Theorem 1.2. For c [, 1], x 1, let Sx, c denote the number of those integers n not exceeding x such that min{p n, P n + 1} > x 1 c. Then, for any fixed c ], 1 5 [ and uniformly for c [, c ], x, we have 1 v 1 2 Sx, c 2x log 1 v 2c 1 v + ox. Remark. Under a suitable strong form of the Elliott Halberstam hypothesis, we get the better bound 1 3 Sx, c x 1 v log 1 v c 1 v + ox. Note that 1 1 follows from 1 2 by selecting ε n = 1 if P n > N 1 c and P n > P n + 1, ε n = 1 if P n + 1 > N 1 c and P n + 1 > P n and ε n =

2 2 R. de la Bretèche, C. Pomerance & G. Tenenbaum in all other cases. Indeed, with these choices for ε n, we obtain that for each prime p > N 1 c N 1/2, the exponent on p in the prime factorization of the rational number AN/BN is n<n P n=p 2 n<n P n+1>p n=p 2 n<n P n 1>P n=p 2. Thus, log AN We have n N P n>n 1 c n N P n,p n+1>n 1 c 2 log P n n N P n,p n+1>n 1 c 2 log min{p n,p n + 1} = 2 log min{p n, P n + 1}. 1 u log N dsn, u { = log N 1 csn, c + } SN, u du, and since the number of n < N with P n > N 1 c is N log1 c+on uniformly for c 1/2, log P n = cn log N + on. We thus obtain n<n P n>n 1 c { log AN 2log N cn 1 csn, c 2Nlog N { } gc + o1, } SN, u du + on where we have set gc := c 1 cfc fu du, with fu := 2 u log 1 v 1 v 2u 1 v. We check by computation that g c = kc. This implies the desired estimate.

3 2. Proof of Theorem 1.2 Products of ratios of consecutive integers 3 We employ the Rosser Iwaniec sieve. A sightly better bound could be obtained from a more sophisticated sieve method, but we do not pursue such improvement here. We refer to [4], [5] for a complete reference of the Rosser-Iwaniec coefficients and merely recall the property we shall use. We denote by γ the Euler constant, and we let p run over primes. Lemma 2.1. Let Q denote a set of primes, let z 2 and write Qz :=, p Q p. There exists a sequence {λ d } d=1 of real numbers, vanishing for d > z or µd =, satisfying λ 1 = 1, λ d 1, and and such that for any number α >, d Qz λ d wd d p Q µ 1 λ 1, 1 wp p {2e γ + O α uniformly for all multiplicative functions w satisfying 1 }, log z 1/3 i ii u<p v, p Q 1 wp p < wp < p p Q, 1 log v 1 + α 2 u v z. log u log u If n is counted by Sx, c, then n = ap 1 = bp 2 1, where p 1 and p 2 are primes greater than x 1 c. Then a and b are obviously coprime, and moreover 2 ab. We need an upper bound for the number Za, b of admissible pairs p 1, p 2 for given a, b. Let C be a sufficiently large constant and set z := x/a 1/2 b 1 log x C. If Q is the set of all primes not dividing a and with {λ d } d=1 the sequence from Lemma Lemma 2.1, we plainly have Za, b p 1 x/a ap 1 1 mod b d Qz λ d µ 1 ap 1 + 1/b, Qz p 1 x/a ap 1 1 mod bd Let us put, for real y 2 and integers q, l with q 1, πy; q, l := p y p l mod q 1. 1, Ey; q := max πy; q, l liy/ϕq. l,q=1

4 4 R. de la Bretèche, C. Pomerance & G. Tenenbaum We apply Lemma 2.1 to the multiplicative function d dϕb/ϕbd. Using the fact that a, bd = 1 for each d Qz, and noticing that c bounded below 1/5 ensures that z b when x is large enough, we deduce that 2 1 Za, b Ma, b + Ra, b with Ra, b := d z Ex/a; bd and Ma, b := p>2 d Qz λ d lix/a ϕbd {2e γ + o1} lix/a ϕb = {2e γ + o1} lix/a b p ab p ab p 1 p p b p 2. p 1 Now we observe that, uniformly as x tends to and a, b vary in the specified ranges, p 2 = 2 pp 2 p 1 p e γ p A log z p>2 where Therefore, writing we obtain that the estimate A := p>2 2 2 Ma, b hn := p n p> pp 2 p 1, p 2 {8 + o1}habx Aab logx/a logx/ab 2 holds uniformly for a x c, b x c, a, b = 1, as x. Let τm denote the number of divisors of m. By the Bombieri Vinogradov theorem, we have, with X a := x/a 1/2 log x C, Ra, b τmex/a; m b x c m X a { } 1/2 Ex/a; m τm 2 Ex/a; m m X a m X a x alog x 2,

5 Products of ratios of consecutive integers 5 where we have used the trivial estimate Ex/a; m x/am and the well-known fact that m x τm2 /m log x 4. Therefore, we obtain from 2 1 and 2 2 Sx, c Za, b 2 3 We have for ν = or where Hs := p>2 1 + a x c, b x c a,b=1, 2 ab 8 + o1 x A a x c b 1 b,a=1 ha a logx/a b x c 2 ab b,a=1 hb x b logx/ab 2 + O log x h2 ν b b s = HsG a sζs Re s > 1 1 p s, G a s := 1 εa p 2 2 s p a p>2. 1 p s 1 + p s /p 2 with εa = 1 if a is even, εa = if a is odd. The functions H and G a can be analytically continued to the half-plane Re s >. Note that H1 = A, G a 1 = 2 εa ha 1. By Selberg Delange estimates see [7], chap. II.5, 2 4 yields in turn and b x c a,b=1, 2 ab b y b,a=1 hb b logx/ab 2 = h2 ν b Ay 2 εa ha y, A 1 4ha log va + o1 x 1 2c v a and v a := log a/ log x. Carrying this back into 2 3, we arrive at Sx, c {2 + o1}x 1 a logx/a log va 1 2c v a = {2 + o1}x 1 a x c, 1 v log 1 2c v 1 v. We remark that with a little more care, the bound 1/5 in the theorem may be replaced with 1/3.

6 6 R. de la Bretèche, C. Pomerance & G. Tenenbaum 3. Further remarks In [2] it is shown that if N is large, than for at least.99n values of n N we have P n > P n + 1, and for at least.99n values of n N we have P n < P n + 1. It follows from Theorem 1.2 that each inequality occurs on a set of integers n of lower asymptotic density 1 log 2 1 c 1 v log 1 v 2c 1 v for each value of c with < c < 1/5. The maximum of this expression is greater than.5544 so we have majorized the result from [2]. Presumably, the set E of integers n with P n > P n + 1 has asymptotic density 1/2. A general theorem of Hildebrand [3] also implies that E has positive lower asymptotic density, but we did not check the numerical value that can be derived from this result. In [2] it is shown that P n < P n + 1 < P n + 2 holds infinitely often, and it was conjectured that so too P n > P n + 1 > P n + 2 holds infinitely often. This conjecture was recently proved by Balog in [1]. We observe that the maximal value AN corresponds to a sequence ε = {ε n } 1 n<n where ε n { 1, 1}. Proposition 3.1. Let N 1. There exists {ε n } 1 n<n { 1, 1} N 1 such that AN BN = 1 n<n n εn. n + 1 Remark. Let A,1 N respectively A 1,1 N, A 1, N the maximum of numerators where the exponents ε n are restricted to {, 1} respectively { 1, 1}, { 1, }. By the proposition, we have A 1,1 N = AN and log A,1 N = 1 2 log AN + Olog N = log A 1,N + Olog N. For example, if {ε n } 1 n<n {, 1} N 1, we have {2ε n 1} 1 n<n { 1, 1} N 1. Since the constant sequence 1 gives the numerator N, we deduce the result. Proof. Take a sequence {ε n } 1 n<n { 1,, 1} N 1 where some ε n =. Write the associated product as A/B with A, B = 1. If we let ε n = 1, the new numerator is A A, n + 1 n B, n, while if we let ε = 1, the new numerator is A A, n n + 1 B, n + 1.

7 Products of ratios of consecutive integers 7 Assuming both of these expressions are smaller than A, we obtain n < A, n + 1B, n and n + 1 < A, nb, n + 1. Multiplying these inequalities and using A, B = n, n + 1 = 1 we obtain nn + 1 < AB, nn + 1, a contradiction. So we may choose ε n {±1} without decreasing the associated numerator. With this method we can replace each value with ±1 and the value of the associated numerator will not decrease. References [1] A. Balog, On triplets with descending largest prime factors, Studia Sci. Math. Hungar , [2] P. Erdős and C. Pomerance, On the largest prime factors of n and n + 1, Aequationes Math , [3] A. Hildebrand, On a conjecture of Balog, Proc. Amer. Math. Soc , no. 4, [4] H. Iwaniec, Rosser s sieve, Acta Arith , [5] H. Iwaniec, A new form of the error term in the linear sieve, Acta Arith , [6] J.-L. Nicolas, Nombres hautement composés, Acta Arith , [7] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge studies in advanced mathematics, no. 46, Cambridge University Press 1995.

8 8 R. de la Bretèche, C. Pomerance & G. Tenenbaum Régis de la Bretèche École Normale Supérieure Département de Mathématiques et Applications 45, rue d Ulm 7523 Paris cedex 5 France Carl Pomerance Lucent Technologies Bell Laboratories 6 Mountain Avenue Room 2C-379 Murray Hill, NJ 7974 USA Gérald Tenenbaum Institut Élie Cartan Université de Nancy 1 BP Vandœuvre Cedex France