Time-Dependent Billiards

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RESEARCH NOTES

ON A DENSITY PROBLEM OF ERDÖS

SAFWAN AKBIK

(Received 13 April 1998 and in revised form 10 June 1998)

Abstract.For a positive integern, letP(n)denotes the largest prime divisor ofnand deﬁne the set:᏿(x)=᏿= {n≤x:ndoes not divideP(n)!}. Paul Erdös has proposed that

|S| =o(x)asx→ ∞, where|S|is the number ofn∈S. This was proved by Ilias Kastanas.

In this paper we will show the stronger result that|S| =O(xe^−1/4 logx).

Keywords and phrases. Number theory, primes, factorial, density, divisibility.

1991 Mathematics Subject Classiﬁcation. 11B05, 11N25.

Introduction. For a positive integern, letP(n)denote the largest prime divisor of nand deﬁne the set

᏿(x)=᏿=

n≤x:ndoes not divideP(n)!}. (1) Paul Erdös [1] proposed that|S| =o(x)asx→ ∞, where|S|is the number ofn∈S. A solution [3] was provided by Ilias Kastanas. There was also [3] a claim of proving that

|S(x)| =O(x/logx). In this paper, we show the stronger result.

Theorem. For some constanta >0, we have

|S| =O xe^−a√

logx

. (2)

In fact,a=1/4suﬃces.

Lemma1. Letν(n)be the number of distinct prime divisors ofn. Deﬁne S1=

n≤x:ν(n) >4kloglogx, k≥1

. (3)

Then

|S1| =O x

(logx)^k

(4) uniformly ink.

Proof. It is well known [2] that if d(m) is the number of divisors of m, then

m≤xd(m)=O(xlogx). Sinced(m)≥2^ν(m), O(xlogx)≥

m≤x

2^ν(m)≥

m∈S1

2^ν(m)≥

m∈S1

(logx)^(4log2)k≥ |S1|(logx)^k+1, (5)

and the lemma follows.

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656 SAFWAN AKBIK

Lemma2. LetC(x)=C=(logx)^k, wherek=k(x)will be chosen later. Deﬁne S2=

n≤x:p²|nfor some primep > C

. (6)

Then

|S2| =O x

(logx)^k

. (7)

Proof. Sincen∈S2if and only ifn=tp²for someC < p≤√xand somet≤x/p²,

|S2| =

C<p≤√ x

x p²

≤

C<p≤√ x

x p²=O

x C

=O x

(logx)^k

. (8)

The ﬁrst big O in (8) follows since

p>C1/p² ≤ _∞

[C]du/u² = 1/[C] = O(1/C), forC≥1.

Lemma3. Let S3=

n≤x:p^α|nfor someα≥Tand some primeP≤C

, (9)

whereT=2logC. Then

|S3| =O x

(logx)^k

. (10)

Proof.

|S3| =

p≤Cα≥T

x p^α

≤2

p≤C

x

p^T ≤2 x 2^T _p

4

p²=Ox 2^T

=O x

C^2log2

≤O x

C

=O x

(logx)^k

.

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The ﬁrst inequality in (11) is valid because

α≥T1/p^α≤1/p^T+1/p^T+1+ ··· ≤2/p^T and the second is valid becausep^T=2^T(p/2)^T≥2^T(p/2)²forT≥2.

Lemma4. LetS(x)=S−(S1∪S2∪S3), then, for any n∈S, we have

P(n)≤2CT . (12)

Proof. Letn∈S. Thenn∈Sand sondoes not divideP(n)!. There exists a prime p0dividingnsuch that

νp0(n) > νp0

P(n)!

, (13)

whereνp(m)denotes the largest integertsuch thatp^tdividesm. Sinceνp₀(P(n)!)≥ 1, (13) implies thatνp0(n)≥2. Sincen∈S2, it follows thatp0≤C. Alson∈S3, so thatT≥νp0(n). Hence,

T≥νp0(n) > νp0

P(n)!

≥P(n)

2p0 ≥P(n)

2C (14)

(3)

which implies (12). Note that the third inequality in (14) is true because νp(m!)≥ m/2pfor anymand anyp|m. This is because

νp(m!)≥ m

p

>m

p −1≥ m

2p, p=m, (15)

and[m/p]=1≥m/2pifp=m.

Proof of the Theorem. For n∈S, we haven∈S1∪S2∪S3. Thus, νp(n)≤ 4kloglogx. Also, ifp^αis any prime power dividingn, then one of the following two possibilities must occur:

(a) p≤Candα≤T, (b) p > Candα=0 or 1.

Case (a) generates at mostC(T+1)≤2CT prime powers. For Case (b), the number of prime powersp^αwithp > C andα≤1 is at mostP(n). By Lemma 4, this is at most 2CT. Hence, the number of possible prime powersp^αthat divide ann∈Sis at most 4CT. But such anncan be the product of at most 4kloglogxdistinct prime powers.

Therefore,

|S| ≤(4CT )^4k^loglogx=(8ClogC)^4k^loglogx

=e^4k^loglog^x(log8+kloglogx+logk+logloglogx)≤e^8k²^(loglogx)² (16) since log8+log(kloglogx)≤kloglogx, forkloglogx≥4.

Choosing

k=1 4·

logx

loglogx, (17)

(16) gives|S| ≤e^(1/2)logx=x^1/2. Hence, S=O

xe^−(1/4)√

logx

. (18)

From (17), we havex/(logx)^k=xe^−1/4√

logx. Lemmas 1, 2, and 3 imply that

|Si| =O xe^−1/4√

logx

, i=1,2,3. (19)

Finally,S=S∪[S∩(S1∪S2∪S3)]. Hence, (18) and (19) yield

|S| ≤ |S|+|S1|+|S2|+|S3| =O

xe^−(1/4)√

logx

, (20)

and (2) follows witha=1/4.

Remark. Ifπ(x)is the number of prime integers that are less than or equal tox, an early version of the prime numbers theorem asserts that

π(x)= _x

2

du logu+O

xe^−a√

logx

, (21)

for some constanta. Although the bigOterms in (19) and (2) are similar, there is no apparent relationship between the PNT and (2).

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658 SAFWAN AKBIK References

[1] P. Erdös,A Proposed Problem, Amer. Math. Monthly98(1991), 965.

[2] G. H. Hardy and E. M. Wright,An Introduction to the Theory of Numbers, 5th ed., The Claren- don Press, Oxford University Press, New York, 1979. MR 81i:10002. Zbl 423.10001.

[3] I. Kastanas,The smallest factorial that is a multiple ofn, Amer. Math. Monthly101(1994), 179.

Akbik: Department of Mathematics, Hofstra University, Hempstead, NY11550, USA

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Special Issue on

Time-Dependent Billiards

Call for Papers

This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.

This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under diﬀerent approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many diﬀerent fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).

We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.

To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.

Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System athttp://

mts.hindawi.com/according to the following timetable:

Manuscript Due March 1, 2009 First Round of Reviews June 1, 2009 Publication Date September 1, 2009

Guest Editors

Edson Denis Leonel,Department of Statistics, Applied Mathematics and Computing, Institute of Geosciences and Exact Sciences, State University of São Paulo at Rio Claro, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil; [email protected]

Alexander Loskutov,Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia;

[email protected]

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