A Note on the Distribution of the Number of Prime Factors of the Integers

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1 A Note o the Distributio of the Number of Prime Factors of the Itegers Aravid Sriivasa 1 Departmet of Computer Sciece ad Istitute for Advaced Computer Studies, Uiversity of Marylad, College Park, MD Abstract The Cheroff-Hoeffdig bouds are fudametal probabilistic tools. A elemetary approach is preseted to obtai a Cheroff-type upper-tail boud for the umber of prime factors of a radom iteger i {1, 2,..., }. The method illustrates tail bouds i egatively-correlated settigs. Key words: Cheroff bouds, probabilistic umber theory, primes, tail bouds, depedet radom variables, radomized algorithms 1 Itroductio Large-deviatio bouds such as the Cheroff-Hoeffdig bouds are of much use i radomized algorithms ad probabilistic aalysis. Hece, it is valuable to uderstad how such bouds ca be exteded to situatios where the coditios of these bouds as preseted i their stadard versios, do ot hold: the most importat such coditio is idepedece. Here, we show oe such A earlier versio of this work appeared i the Proc. Hawaii Iteratioal Coferece o Statistics, Mathematics, ad Related Fields, address: sri@cs.umd.edu Aravid Sriivasa). URL: sri Aravid Sriivasa). 1 This research was doe i parts at: i) Corell Uiversity, Ithaca, NY supported i part by a IBM Graduate Fellowship), ii) Istitute for Advaced Study, Priceto, NJ supported i part by grat of the Alfred P. Sloa Foudatio), iii) DIMACS Ceter, Rutgers Uiversity, Piscataway, NJ supported i part by NSF-STC ad by support from the N.J. Commissio o Sciece ad Techology), ad iv) the Uiversity of Marylad supported i part by NSF Award CCR ). Preprit submitted to Elsevier 17 October 2008

2 extesio, to the classical problem of the distributio of the umber of prime factors of itegers. For ay positive iteger N, let νn) deote the umber of prime factors of N, igorig multiplicities. Let l x deote the atural logarithm of x, as usual. It is kow that the average value of νi), for i [] = {1, 2,..., }, is µ. = l l + O1) ± Ol 2 ), for sufficietly large. See also the discussio i Alo & Specer [1]. We are iterested i seeig if there is a sigificat fractio of itegers i [] for which νi) deviates largely from µ. Formally, Hardy & Ramauja [4] showed that for ay fuctio ω) with lim w) =, {i [] : νi) l l + ω) l l } = o1), 1) where the o1) term goes to zero as icreases. Their proof was fairly complicated. Turá [9] gave a very elegat ad short proof of this result; his proof is as follows. Let E[Z] ad V ar[z] deote the expected value ad variace of radom variable Z, respectively. Defie P to be the set of primes i []. For a radomly picked x [], defie, for every prime p P, X p to be 1 if p divides x, ad 0 otherwise. Clearly, νx) = p P X p. Hece, µ = E[νx)] = E[X p ] = /p p P p P ad thus, µ µ. = 1 p P p = l l ± Ol 2 ), where the last equality follows from Mertes theorem see, for istace, Rosser & Schoefeld [7]). By Chebyshev s iequality, P r νx) µ) λ) V ar[νx)] λ 2 ad by obtaiig good upper bouds o the variaces V ar[x p ] ad the co- 2

3 variaces Cov[X p, X q ], Turá obtais his result that ) l l P r νx) µ λ) O, 2) λ 2 which, i particular, implies 1). Erdős & Kac [3] show that as, the tail of νx) ad of ay fuctio from a fairly broad class of fuctios of x) approaches that of the correspodig ormal distributio, i.e., that if ω is real ad if K = {i [] : νi) l l + ω 2 l l }, the K lim = π 1/2 w e u2 du. 3) We stregthe the upper-tail part of 2) by showig that for ay ad ay parameter δ > 0, e δ ) µ P rνx) µ 1 + δ)). 1 + δ) 1+δ I cotrast with 3), we get a boud for every ; thus, for istace, we get a cocrete boud for deviatios that are of a order of magitude more tha the stadard deviatio. We poit out that strog upper- ad lower-tail bouds are kow usig o-probabilistic methods [6]. The goal of this ote is to show that a simple probabilistic approach suffices to derive expoetial upper-tail bouds here. We also hope that the method ad result may be of pedagogic use i showig the stregth of probabilistic methods, ad i the study of tail bouds for egatively) correlated radom variables. 2 Large Deviatio Bouds We first quickly review some saliet features of the work of Schmidt, Siegel & Sriivasa [8]. 2.1 Cheroff-Hoeffdig type bouds i o-idepedet scearios The basic idea used i the Cheroff Hoeffdig heceforth CH) bouds is as follows [2,5]. Give radom variables heceforth r.v. s) X 1, X 2,..., X, 3

4 we wat to upper boud the upper tail probability P rx a), where X. = i=1 X i, µ. = E[X], a = µ1 + δ) ad δ > 0. For ay fixed t > 0, P rx a) = P re tx e at ) E[etX ] e at ; by computig a upper boud ut) o E[e tx ] ad miimizig ut) e at over t > 0, we ca upper boud P rx a). Suppose X i {0, 1} for each i, a commoly occurig case. I this case, a commoly used such boud is P rx µ1 + δ)) F µ, δ) =. e δ ) µ 4) 1 + δ) 1+δ see, for example, [1]). Oe basic idea of [8] whe X i {0, 1} is as follows. Suppose we defie, for z = z 1, z 2,..., z ) R, a family of fuctios S j z), j = 0, 1,...,, where S 0 z) 1, ad for 1 j, S j z). = 1 i 1 <i 2 <i j z i1 z i2 z ij. The, for ay t > 0, there exist o-egative reals a 0, a 1,..., a such that e tx i=0 a i S i X 1, X 2,..., X ). So, we may cosider fuctios of the form y i S i X 1, X 2,..., X ) i=0 where y 0, y 1,..., y 0, istead of restrictig ourselves to those of the form e tx, for some t > 0. For ay y = y 0, y 1,..., y ) R +1 +, defie f y X 1, X 2,..., X ) =. i=0 y i S i X 1, X 2,..., X ). The, it is easy to see that )) a a P rx a) = P r f y X 1,..., X ) y i i=0 i E[f yx 1,..., X )] ai=0 y i a i). So, the goal ow is to miimize this upper boud over y 0, y 1,... y ) R Assumig that the X i s are idepedet, it is show i [8] that the optimum for the upper tail occurs roughly whe: y i = 1 if i = µδ, ad y i = 0 otherwise. We ca summarize this discussio by Theorem 2.1 [8]) Let bits X 1, X 2,... X be radom with X = i X i, ad 4

5 let µ = E[X], k = µδ. The for ay δ > 0, P rx µ1 + δ)) E[S kx 1, X 2,..., X )] ). µ1+δ) k If the X i s are idepedet, the this is at most e δ ) µ. 1 + δ) 1+δ 2.2 Tail Bouds for νx) Returig to our origial sceario, let be our give iteger. For a radomly picked x [], let X p be 1 if p divides x, ad 0 otherwise. As stated earlier, νx) = p P X p. Let { ˆX p p P } be a set of idepedet biary radom variables with P rx p = 1) = 1/p. For ay r ad ay set of primes p i1, p i2,..., p ir, ote that r r E[ X pij ] = P r p ij x)) j=1 j=1 = / r j=1 p ij 1 rj=1 5) p ij = E[ r j=1 ˆX pij ]. Thus we get Theorem 2.2 For ay 2 ad for ay δ > 0, e δ ) µ P rνx) µ 1 + δ)), 1 + δ) 1+δ just by ivokig Theorem

6 3 Variats Why does our approach ot work directly for the lower tail of νx) also? The reaso is that a direct egative-correlatio result aalogous to 5) does ot appear to hold. It would be iterestig to see if good lower-tail bouds also ca be obtaied by a short proof; as i [3,9], it may be possible to make quatitative use of the fact that the {X p } are all almost idepedet. It is kow that coutig the prime divisors icludig multiplicity chages the fuctios a little [6], ad it would be worth cosiderig short probabilistic) proofs for the tail behavior of this fuctio also. More geerally, ca we cocretely exploit the ear-idepedece properties of additive umber-theoretic fuctios [3]? Ackowledgmets. I thak Noga Alo, Eric Bach, Carl Pomerace, Christia Scheideler ad Joel Specer for valuable discussios & suggestios. Refereces [1] N. Alo ad J. Specer. The Probabilistic Method, Secod Editio. Joh Wiley & Sos, Ic., [2] H. Cheroff. A measure of asymptotic efficiecy for tests of a hypothesis based o the sum of observatios. Aals of Mathematical Statistics, 23, ). [3] P. Erdős ad M. Kac. The Gaussia law of errors i the theory of additive umber theoretic fuctios. America Joural of Mathematics, 62, ). [4] G. H. Hardy ad S. Ramauja. The ormal umber of prime factors of a umber. Quarterly J. Math., 48, ). [5] W. Hoeffdig. Probability iequalities for sums of bouded radom variables. America Statistical Associatio Joural, 58, ). [6] C. Pomerace. O the distributio of roud umbers. Number Theory Proceedigs, Ootacamud, Idia, K. Alladi, ed., Lecture Notes i Math. 1122, ). [7] J. B. Rosser ad L. Schoefeld. Approximate formulas for some fuctios of prime umbers. Illiois Joural of Mathematics, 6, ). [8] J. P. Schmidt, A. Siegel, ad A. Sriivasa. Cheroff-Hoeffdig bouds for applicatios with limited idepedece. SIAM Joural o Discrete Mathematics, 8, ). [9] P. Turá. O a theorem of Hardy ad Ramauja. Joural of the Lodo Mathematics Society, 9, ). 6

Asymptotic distribution of products of sums of independent random variables

Asymptotic distribution of products of sums of independent random variables Proc. Idia Acad. Sci. Math. Sci. Vol. 3, No., May 03, pp. 83 9. c Idia Academy of Scieces Asymptotic distributio of products of sums of idepedet radom variables YANLING WANG, SUXIA YAO ad HONGXIA DU ollege