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Explicit Estimates Involving the Primorial Integers and Applications

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1 2 Journal of Integer Sequences, Vol. 24 (2021), 3 47 6 Article 21.7.8 23 11 Explicit Estimates Involving the Primorial Integers and Applications Safia Aoudjit and Djamel Berkane University of Blida Department of Mathematics P. O. Box 270, Route de Soumaa Blida Algeria [email protected] [email protected] Pierre Dusart XLIM UMR 7252 Facult´edes sciences et techniques Universit´ede Limoges P. O. Box 123, avenue Albert Thomas 87060 Limoges Cedex France [email protected] Abstract In this paper, we propose several explicit bounds for the function that counts the number of primorial integers less than or equal to a given positive real number. As ap- plications, we obtain an effective version of P´osa'sinequality, and a method to estimate P the maximal value of the sum over prime divisors pjq f(p) for a positive decreasing function f, when q ranges over all integers less than x. In particular, we improve the P log p upper bound for the maximal value of pjq p−1 . 1 1 Introduction and statement of results As usual, let (pk)k≥1 denote the increasing sequence of prime numbers, and let Nk be the primorial (prime-factorial) integer of index k, the product of its k first terms. The integers Nk are the terms of the sequence A002110 in the On-line Encyclopedia of Integer Sequences (OEIS) [22], and play an important role in number theory from Euclid's proof of the infinity of primes to the remarkable equivalences of the Riemann hypothesis due to Nicolas [13] and Robin [19]. The approximations linked to Nk have found some unexpected applications in various areas of number theory; see the papers of Betts [5], Planat et al. [16] and Zhang [23]. However, the most frequently used results concern their logarithm θ(pk) where θ denotes the Chebyshev function, here we cite the recent papers of Axler [1] and Dusart [8]. The principal object of this manuscript is to study the function K(x) which counts the number of primorial integers less than or equal to x; for instance K(1) = 0, K(2) = 1, K(3) = 1, K(5) = 1, K(6) = 2,. and a table of some large values of x is given in Section7. The function K(x) is very close to π(x) (the usual prime counting function) since it will be shown in what follows that K(x) ≈ π(log x). It often appears implicitly as an important key in the demonstration of several results; see, for instance, Balazard [2] and Hassani [11]. We let logi denote the i-fold iterated logarithm. The asymptotic expansion of K(x) is easy to evaluate. Let us show this using less precise results than those in Balazard [2]: Theorem 1 (Balazard). For every real number x ≥ 2 and every integer m ≥ 0, we have m ! log x X j! 1 K(x) = + O : log x j logm+1 x 2 j=0 log2 x 2 Proof. Since for a real x ≥ 2, the integer K := K(x) is also given by the inequalities θ(pK ) ≤ log x < θ(pK+1); using the prime number theorem in the forms t θ(t) = t + O m logm+2 t and t π(t) = li(t) + O ; m logm+2 t where li(x) indicates the usual logarithmic integral function, one gets successively log x pK = log x + Om m+2 ; log2 x log x K(x) = π(pK ) = li(log x) + Om m+2 : log2 x 2 Finally, by substituting log x in the classic asymptotic expansion of li(x) below Z x dt x 1! m! 1 li(x) = = 0! + + ··· + m + O m+1 ; 2 log t log x log x log x log x we establish the result. Consequently, the asymptotic expansion of the function xK(x) corresponds to a type of formula among a large class of expansions related to the iterated logarithms studied in Belbachir and Berkane [4] to which an asymptotic expansion for the sum of inverses is determined. The following proposition is a straightforward application of Belbachir and Berkane [4, Theorem 1]: Proposition 2. For every real number x ≥ 2 and every integer m ≥ 2, we have X 1 1 = log2 x − log x − log x + C nK(n) 2 2 2 3 2≤n≤x δ2 δm 1 + + ··· + m−1 + O m ; log2 x (m − 1) log2 x log2 x where C is an absolute constant, and fδjgj≥0 is the sequence A233824 in the OEIS [22] given by the recurrence relation δn + 1!δn−1 + 2!δn−2 + ··· + (n − 1)!δ1 = n · n!. In this article, by comparing K(x) with π(log x) over large and special ranges, our first result (see Theorem 10) is an effective version of an inequality of P´osa[15], where he proved n that for all n > 1 there is a kn such that Nk > pk+1 for all n ≥ kn. We describe an algorithm calculate kn for every n. The second result (see Theorem 19) is a good improvement of an approximation of the P log p greatest sum over prime divisors L(q) = pjq p−1 when q ranges over all integers not ex- ceeding x. The method employed to prove this theorem can also be applied to evaluate the P maximum of sum of type Lf (q) = pjq f(p) for any positive decreasing function f on (1; 1). In Section5, we show that the maximal value of Lf (q) is Lf (NK(x)). These results are follow-ups of fully explicit bounds for the O-term in the formula of K(x), mainly Theorems 14{18. Finally, throughout this paper, e represents Napier's constant, p a prime number and the calculations are made by using Maple 17. 2 Technical lemmas log Nk The sequence log pk has the same asymptotic behavior as its Ces`aroaverage k , where the first few terms are log N log k − 2 log2 k − 6 log k + 11 log3 k k = log k + log k − 1 + 2 − 2 2 + O 2 : k 2 log k 2 log2 k log3 k 3 Robin [20] and Dusart [7] obtained fully explicit bounds for pk and θ(pk), we gathered the ones we need in the following lemma. Lemma 3. θ(pk) > k log k; 8k ≥ 3; (1) θ(pk) ≥ k(log k + log2 k − a); 8k ≥ 2; and a = 1:0769; (2) log k − 2:1454 θ(p ) ≥ k(log k + log k − 1 + 2 ); 8k ≥ 3; (3) k 2 log k θ(pk) ≤ k(log k + log2 k − 0:9465); 8k ≥ 14; (4) 1 p ≤ k(log k + log k − ); 8k ≥ 20; (5) k 2 2 pk ≤ k log pk; 8k ≥ 4: (6) Similarly, the following lemma directly gives an upper bound of θ(pk+1) in terms of k instead of k + 1. Note that c = 1 − log 2 is an absolute constant. Lemma 4. We have, when k ≥ 2: log k + c θ(p ) ≤ k log k + log k − c + 2 : (7) k+1 2 log k Proof. From inequality (5), by taking the logarithm and using the fact that log(1 + x) ≤ x for all x > 0, we easily obtain log k − 0:5 log p ≤ log k + log k + 2 ; (8) k 2 log k this is valid even for k ≥ 18. In particular, replacing k by 2m − 1 and using the inequality log(2m − 1) < log(m) + log(2), one gets that for all m ≥ 2: log m + log 2 − 0:5 log 2 log p ≤ log m + log m + log 2 + 2 + : 2m−1 2 log m log2 m Since the right-hand side is a strictly increasing function M(m), the sum of all log p2m−1 R k until k − 1 is bounded above by 2 M(t)dt, and we have k−1 X log k + c log p ≤ k log k + log k − c + 2 − D(k); 2m−1 2 log k m=1 where 3 3 log 2 2 D(k) = 2 log 2 − li(k) + − 2 log 2 li(2) − 3 log 2 − 2 log 2 + 4 − 2 2 − : 2 2 2 log 2 log 2 Finally, since the function D(x) is negative when x ≥ 11, by applying the logarithm to the inequality obtained in the lemma below, one gets the result for k ≥ 11. A computer check handles the cases 2 ≤ k ≤ 11. 4 Lemma 5. We have, when k ≥ 11: k−1 Y Nk+1 < p2i−1: i=1 Proof. We proceed by induction on k. The inequality holds for k = 11. As k ≥ 3 implies 2k − 1 ≥ k + 2 and then pk+2 ≤ p2k−1, the lemma follows from the fact that k k+1 Y Y Nk+2 < pk+2 p2i−1 ≤ p2i−1: i=1 i=1 Lemma 6. For all k ≥ 1, we have 5 p ≤ p ; k+1 3 k and the inequality is sharp at k = 2. 1 Proof. According to Dusart [7], the interval [x; 25 log2(x) ] contains at least one prime for all x ≥ 396738. As 396833 is prime, we have that, for pk ≥ 396833 = p33609, 1 4001 5 pk+1 ≤ (1 + 2 )pk < pk < pk: 25 log (pk) 4000 3 Finally, by computer, the last inequality is shown to be also valid for 2 ≤ pk ≤ 396832. Lemma 7. For every positive integer m, we have m Y Nk ≤ N m(m+1) : 2 k=1 Pm Qm Proof. By setting s := k=1 k, we want to show that k=1 Nk is less than the primorial of index s.
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