Lune, J. (Jan) van de

English

CWI's Institutional Repository

STICHTING t1/\THEMATISCH CENTRUM 2e Boerhaavestraat 49 AMSTERDAM AFDELING ZUIVERE WISKUNDE zw 1969-004 On the distribution of a specific number-theoretical sequence by J. van de Lune december 1969 MATHEMArlSOi PJt.ln.te.d at .the. Ma:tkema.tic.al Ce.nt/1.e., 49, 2e. Boelthaavu.vcaat, w.teJr.dam. The. Mathematic.al Ce.ntli.e., 6ou.nde.d .the. 11-.th 06 Fe.bJWtVLy 1946, .u a. non-pJr.Of,.lt .i.n.4:U:tu.tlon a..iJn.i.ng at .the. pJr.Omouon 06 pu11.e. mathema.t.lu and U.6 a.ppU.c.a.tio~. U; .u 4po~oll.e.d by .the. Ne..thelLi.a.ncl.4 Gove/1.nme.n.t -thltou.gh .the. Ne..the.Jt.t.a.nd.4 OJiga.nlza.tion 60Jt. .the. Adva.nc.eme.nt 06 PWLe Rue.all.Ch (Z.111.0), by .the. Mu.n,lclpaU.ty 06 lw4t.eJu:la.m, by .the. UnlveJUUy 06 Arnllt.eJu:la.m, by .the. F11.e.e. UnlveJUUy at lw4.teltdam, and by .lnchwtluu. On the distribution of a specific number-theoretical sequence by J. van de Lune Introduetion This note must be considered as a continuation of [1], from which we recall some definitions and theorems. For a natural number m .::_ 2 we define g(m) as the largest prime dividing m, whereas g( 1) = 1. We also write g instead of g(m), m Let G(n,a) be the number of natural numbers m with the properties m < n and g(m) .::_ ma, where a is a fixed real number. It can be shown that the function G(a) d.¥,f . 1 G(n,a) lim -n n➔oo lS continuous and satisfies: G(a) = o, G(a) = 1 ' G' ( a) = J_ G(~) a 1-a ' Defining i H(x) = H(x) = G(J_) X (a.::_0) ( a.::_ 1 ) (O<a.<1). for O < x < for x > 1, it is easy to see that H(x) is continuous on x > 0 and satisfies the equation H' (x) - - J_ H(x-1), X ( x> 1 ) • From this it follows that xH(x) = Ix H(t)dt, x-1 2 ( x~ 1 ) , and by means of this formula it is easily shown that H(x) is a positive function which tends to zero very rapidly when x tends to infinity. We now define the sequence ,\k, (k= 1 ,2, 3, ... ) as follows (k=2,3,4, ... ). It is to be expected that ,\k behaves very irregular and while tabulating this sequence one might conjecture for example that the sequence Ak is uniformly distributed modulo 1 • However, in this note it will be shown that the sequence Ak is not uni-formly distributed modulo a, for any positive a. 1. Lemma 1. If the function f(x) is such that the integral fA1 f(x)dH(x), (A> 1) exists a.s an ordinary Riemann,..Stieltjes integral then . 1 lim -n n-+co l f(.\k) = - IA f(x)dH(x). k<n 1 ,\ <A k Proof. On the interval [O,A] we construct a subdivision 1 = a0 < a 1 < a < < a < a = A and we define 2 • · • m-1 m M = \) m = \) sup a <x<a v-1- - v inf a <x<a v-1- - v f(x), f(x). 3 Since ( f(x)dll(x) exists we may choose the sub di vision of [ 1 ,A] such that m < - IA f(x)dH(x) I M {H(a 1) - H(a )} + £ v=1 \) v- \) 1 and m > - I: f(x)dH(x) - c, I m {H(a 1) - H(a )} v=1 \) v- \) We now write 1 m I f(\) I 'i' f( Ak) = - l n k<n n v=1 a <A <a A <A v-1- k 'v k 'R'<rI and observe that m m I I f( Ak) < I I M = n v=1 a <>-, <1:1 -n v=1 a <A <a \) v-1- k v . v-1- k V k<n k<n 1 m 1 1 = - 'i' M • {G(n, --) - G(n, -)}, n l v a a v=1 v-1 v because of the fact that for all v the number of natural numbers k satisfying the conditions k .::_ n and av_ 1 .::_ Ak < av is equal to 1 1 G(n, -a-) - G(n, -) . v-1 av Since 1 m 1 lim - L M {G(n, --) n-+oo n v=1 v av-1 m M {G(-· -1 -) G(-1 ) } = I v av-1 a v=1 \) m = I M {H(a 1) - H(a )} v=1 V V- V 1 G(n, -)} = a \) = < - I: f(x)dH(x) + c we obtain that • 1 lim sup -n n➔oo 4 In a similar way one also proves that 1 . . 1 im inf -n n➔oo l f(Ak) > - JA f(x)d.H(x) k<n 1 A <A k - e:. Since this is true for all e: > 0 we may conclude: • 1 lim -n n➔oo JA f(x)d.H(x). 1 Theorem. If the function f(x) is such that if(x)I < M for all x > 1 and the integral I: f(x)d.H(x) exists, then lim l I f(Ak) = - J 00 f(x)d.H(x). n➔oo n k=1 1 Proof. We write 1 n - l f(Ak) n k=1 = l l f(Ak) + _n1 n k<n A <A k and fix A> 1 such that M.H(A) is small. Then we have 11 I f(Ak) + J001 f(x)dH(x)I n k=1 < I¾ l f(Ak) + JA1 f(x)d.H(x) I + k<n A <A k +l n I k<n A >A k-5 According to lemma 1 the first of these terms can be made arbitrarily small by taking n large enough. The second is M 1 < - G(n -) - n ' A which tends to M.H(A) as n ➔ 00 , whereas the third term'is < M. J: dH(x) = M H(A). From this it is clear that n Joo lim ¾ L f(Ak) = - f(x)dH(x). n➔00 k=1 1 As an application of this theorem we prove the assertion concerning the distribution of Ak made in the introduction. Let a be any fixed positive number and define the set Et for O < t < a as follows: 00 E = t u r=O {x E ffi; ra 2-_ x 2-_ ra+t} and let ft be the characteristic function of Et. It is easily seen that this ft satisfies the conditions of theorem 1. Thus ~ Ira+t l dH(x) r=O 00 - - I r=O {H(ra+t) - H(ra)}. ra It is rather easy to convince oneself that D(t) is differentiable on the interval 1 - a,[.l] < t < a, a such that D' (t) 00 I 1 r=[-J a 6 00 H' (ra+t) = L r=[.l] a 1 ra+ 1 H(ra+t-1), from which it is obvious that D'(t) is decreasing. Hence D'(t) is not constant. However, from the definition of D(t9 and the assumption that the sequence Ak is uniformly distributed modulo a it would follow that D(t) and D' ( t) t = -a 1 = - = constant. a ( O.s_t.s_a) Conclusion. The sequence Ak is not uniformly distributed modulo a, for any a> O. 2. In this section we will make a few remarks on the behaviour of where f is not bounded. If we take f(x) = 2'2:£. one has and for n = 2m > -n f(A ) n .l f(A ) = -1 f(m) = 2m n n 2m and hence it follows that 7 is divergent. On the other hand, it is easy to show that exists. Thus, it may happen that (°° f(x)dH(x) J 1 exists whereas is divergent. A somewhat more precise result is the next theorem Theorem. If f(x) ,:_ 0 on x 2:_ x0 and f(x) is bounded on each finite interval and I: f(x)dH(x) exists then lim infl I f(Ak) .:_- J00 f(x)dH(x). n-+oo n k=1 1 Proof. For any A 2:_ x0 one has n I f(Ak) > I f(Ak), n k=1 -n k<n A <A ·k so that . . 1 n IA lim inf - l f(Ak) ,:_ - 1 f(x)dH(x) n-+oo n k=1 for each A,:_ x0 , and the theorem follows. 8 A curious consequence of this theorem is, taking f(x) = x, lim inf 1 I A > - I~ x dH(x) n-+oo n k=1 k - 1 --I: x dH(x) = I: H(x)dx = ey = 1,781 .•. , where y is Euler's constant ( c. f ,, [ 1], page 24). Reference: [1] J, van de Lune and E. Wattel, On the frequency of natural numbers m whose prime divisors are all smaller than ma, Mathematical Centre, Amsterdam, Report ZW 1968-007 (1968). 

On the distribution of a specific number theoretical sequence

Abstract

Similar works

Full text

Available Versions

CWI's Institutional Repository