Number of Prime Divisors of. k (n), where. k Is the k-fold Iterate of.

Transcription

1 journal of number theory 65, (997) article no. NT97264 Number of Prime Divisors of. k (n), where. k Is the k-fold Iterate of. N. L. Bassily* Ain Shams University, Abbassia, Cairo, Egypt I. Ka tai - Eo tvo s Lora nd University, H-088, Budapest, Mu zeum krt. 68, Hungary and M. Wijsmuller La Salle University, Philadelphia, Pennsylvania 94 Communicated by A. Granville Received April 30, 996; revised October 8, 996 In this paper we obtain the distribution of the function (. k (n)) which counts the number of distinct prime factors of the k-fold iterate of the Euler. function. With coefficients a k =(k+)! and b k =-(2k+) (k!) we prove that lim x x { } > (. k(n))&a k (log log x) k+ <z b k (log log x) = = k+2-2? Z e &t22 dt & An analogous result is obtained for (. k ( p&)). This extends the results known through the work of P. Erdo s and C. Pomerance, M. R. Murty and V. K. Murty, and I. Ka tai. 997 Academic Press. INTRODUCTION Let P be the set of primes and p, q, p i, q i # P. The function (n) counts the number of distinct prime divisors of n,.(n) represents the Euler-totient function, and (n) is a completely multiplicative function defined on the set of primes by ( p)=p&. The iterates of. and are denoted by. k (n)=. k& (.(n)) and k (n)= k& ((n)) with. 0 (n)= 0 (n)=n. The * Supported in part by the Associate program of ICTP. - Supported in part by OTKA X Copyright 997 by Academic Press All rights of reproduction in any form reserved. 226

2 PRIME DIVISORS OF. k (n) 227 letter c denotes a constant not necessarily the same each time, the letters C, C 2, and C refer to constants which, once chosen, stay the same throughout the remainder of the paper; n always stands for a positive integer. The main purpose of this paper is to prove the following theorems Theorem. For each fixed integer k, let a k =(k+)! and b k =-2k+ (k!). Then for each real number z lim x x > { } (. k(n))&a k (log log x) k+ <z (.) b k (log log x) k+2 ==8(z) where 8 is the standard Gaussian law. Earlier Erdo s and Pomerance [4] and Murty and Murty [9] proved (.) for k= while Ka tai [6] proved (.) for k=2. Theorem 2. For each fixed integer k and a k, b k as in Theorem lim x?(x) > { } (. k( p&))&a k (log log x) k+ b k (log log x) k+2 <z ==8(z) (.2) for every real number z. For k= this is a known result proved by Ka tai in [7]. It is not too difficult to see that (. k (n)) can be approximated by ( k (n)) and this is the technique used in [4]. But for k2 the situation is much more complicated and we introduce a strongly additive function { k (n) which is defined recursively by { 0 ( p)=, { k (p)= q p& { k& (q) For fixed k we apply the well-known results by Kubilius and Shapiro, and Barban, Levin, and Vinogradov, and establish (.) and (.2) for { k (}). Using results from sieve theory we show that ( k (n)) and, therefore, also (. k (n)) can be approximated by { k (n) which leads to (.) and (.2) for (. k (n)). In Section 6 we outline why and how these results also hold when k=k(x) tends to infinity as a function of x. To establish how fast k can grow as a function of x we explicitly display the dependence on k in all error terms.

3 228 BASSILY, KA TAI, AND WIJSMULLER 2. PRELIMINARY LEMMAS From the definition of (n) it is clear that ((n)) (.(n)) (n)+ ((n)). The generalization of this statement can be formulated as Lemma 2.. For all k=0,,2,...we have ( k (n)) (. k (n)) (n)+ ((n))+ } } } + ( k (n)). (2.) Proof. The left-hand side of (2.) is immediate. The right-hand side follows if we show that p. k (n) O p j (n) for some j=0,, 2,..., k (*) which we will do by induction. Clearly (*) holds for k= and we assume that (*) holds for k&. If p. k (n) then p. k& (n) orp (. k& (n)). If p. k& (n) we may apply (*) and p j (n) for some j=0,,2,...,k&. If p %. k& (n) then there is some prime q which divides. k& (n) and p (q). By induction q j (n) for some j=0,,..., k& and, therefore, p j (n) for some j=, 2,..., k which proves (*) and the lemma. Lemma 2.2. Let m be a nonnegative integer and $ a real number with 0<$2. Then there is a number c depending upon m but not upon $ so that the inequality x &$ < holds for all sufficiently large values of x. p m& (?(x, p,)) m $c \ x m log x+ This if a variation of Lemma 4.9 in [2] and can be proved in the same way. Another modification which can be stated as x 2 <(pq)x pq m& (?(x, pq, )) m c \ x m log log x (2.2) log x+ is due to Tenenbaum [0]. We also make repeated use of the BombieriVinogradov mean value theorem which is stated in the following lemma.

4 PRIME DIVISORS OF. k (n) 229 Lemma 2.3. that For every positive constant B there exists a constant A such k- x(log x) A max max?(z, k, l)&li(z) (l, k)= zx }.(k)} << x (log x) B. The value 4B+40 is an appropriate choice for A. For the proof see []. The next lemma can be found as Theorem 3.8 in [5]. Lemma 2.4. For (l, k)= and lk<x 3x?(x, k, l)<.(k) log(xk) holds for all x. Lemma 2.5. For integers l and k let $(x, k, l) = p#l(mod k) Then for l= or &, kx, and x3 we have p. $(x, k, l) C log log x,.(k) where C is an absolute constant. For l=, this is Eq. (3.) in [3] and a proof can be found in [6]. It will follow from the following lemma that { k ( p)b(log log x) k for almost all primes when B is sufficiently large. Lemma 2.6. For H0, x3 let T k (x, H )=>[ { k (p)>h k ]. Then there exists an absolute constant C 2 such that T k (x, H ) Ck 2x(log log x)k&. (2.3) 2 H Proof. If follows from the definition of { k (n) that { ( p)= (p&). From the well-known inequality 2 ( p&) <cx

5 230 BASSILY, KA TAI, AND WIJSMULLER we obtain that T (x, H )= { (p)>h < 2 2 (p&) < cx H 2 H. Thus (2.3) is true for k=. Since { k ( p)= q p& { k& (q) for k, { k (p)>h k implies that either ( p&)>h or { k& (q)>h k& for at least one prime q which divides ( p&). Therefore, T k (x, H )T (x, H)+ qx { k& (q)>h k&?(x, q,) Using partial summation and induction on k as well as Lemma 2.4 it follows that with u=[log xlog 2]& T k (x, H ) cx 2 + H x2<qx { k& (q)>h k& u + j=?(x, q, ) x2 j+ <qx2 j { k (q)>h k& cx k& 2 x(log log x) k&2 u +C + H 2 2 H j= 6(2 j+ ) j(log 2) k&\ T x 2 j, H + cx k& 2 x(log log x) k&2 +C + 24x(C 2 log log x) k& H 2 2 H 2 H Ck 2x(log log x)k& 2 H when C 2 =max(c, 26). For all primes p, { 0 ( p)2 log p and therefore, by induction, for all k, { k ( p)2 log p which means that { k (n)2 log n. When one applies Lemma 2.6 with H0(log log x) it follows that qx { k (q)>h k holds for j=0,, 2, and 3. { k(q) j q =O \ (C 2 log log x) k (2.4) log 2 x + 3. THE MOMENTS OF { k (n) We choose a constant C such that C=max(C, C 2 ). For a fixed k let S k (x) = { k (p), A k (x) = { k (p) p.

6 PRIME DIVISORS OF. k (n) 23 Lemma 3.. For every k=, 2,..., we have S k (x)=li(x) A k& (x)+o(li(x)(c log log x) k& ) (3.) as x. Proof. Let H=C log log x in Lemma 2.6. To simplify notation we denote a summation over primes q for which { k& (q)h k& by $. It follows from (2.4) that S k (x)= = $ qx q p& { k& (q) { k& (q)?(x, q,)+o \(Clog log x)k& x log 2 x +. We split the sum over q into two parts, $ when q<x 3 and $ 2 when x 3 qx. It follows from Lemma 2.2 and 2.3 that $ { k&(q) =li(x) $ q& +O \ (C log log x)k& x log 2 x +. q<x 3 From the choice of H and (2.4) it follows that { k&(q) $ q& =A k&(x)+o((c log log x) k& ). q<x 3 Furthermore, by Lemmas 2.2 and 2.4 if follows that $ H k& 2 x 3 qx and this completes the proof of (3.). x?(x, q,)<<(c log log x) k& log x Lemma 3.2. For every k=0,, 2,..., as x A k (x)= (k+)! (log log x)k+ +O((C log log x) k ). (3.2) Proof. When k=0 Eq. (3.2) is clearly true. We assume that (3.2) holds for k& and use induction on k. It follows from (3.) that S k (x)=li(x) (log log x)k k! +O(li(x)(C log log x) k& ).

7 232 BASSILY, KA TAI, AND WIJSMULLER Using partial summation, and the proof is complete. A k (x)= S k(x) x + x S k (u) du+o() x 0 u 2 (log log x)k+ = +O((C log log x) k ) (k+)! Lemma 3.3. For any k=,2,3,...let D k (x) = { 2 k (p). Then D k (x)=li(x) (log log x)2k (k!) 2 +O(li(x)(C log log x) 2k&2 ). (3.3) Proof. It follows from Lemma 2.6 and partial summation that D k (x)<<li(x)(c log log x) 2k. Let { k ( p)= q p& { k& (q)= + = f (p)+f 2 (p); q<x 6 qx 6 q p& q p& then U(x) = f 2 (p)d k(x) =U(x)+ =U(x)+ +. f (p) f 2 (p)+ 2 f 2 2 (p) Using Lemmas 2.4 and 2.6, as well as Lemma 2.2 and its generalization stated in (2.2), we find that 2 { 2 k& (q)?(x, q,)+ { k& (q ) { k& (q 2 )?(x, q q 2,) qx 6 q,q 2 x 6 Rli(x)(C log log x) 2k&.

8 PRIME DIVISORS OF. k (n) 233 Furthermore, by the CauchySchwartz inequality which means that (U(x)) 2 It follows from Lemma 2.3 that 2 2 \ D 2+ k (x) \ 2 2+ =O(li(x)(C log log x) 2k&2 ). U(x)= { k& (q ) { k& (q 2 )?(x, q q 2,)+ q {q 2 q, q 2 <x 6 =li(x) A 2 k&(x 6 )+O(li(x)(C log log x) 2k& ). q<x 6 { 2 k& (q)?(x, q,) Since A k& (x)&a k& (x 6 )=O((C log log x) k& ) (3.3) follows immediately. Lemma 3.4. For every k=0,,2,...let { 2 B 2 k (x) = (p) k p. Then log B 2 x)2k+ log log x)2k+2 k (x)=(log +O (2k+)(k!) 2 \(C 2k+2 +. (3.4) Proof. Induction on k and Lemma 3.3 will give the desired result. 4. THE LIMIT DISTRIBUTION OF { k (n) In [8] Kubilius defines a special class of functions which denotes by the symbol H. It follows from Lemma.5 in [2] that a strongly additive function f(n) belongs to the class H if (i) B(x) = \ f 2 2 (p) (x) p + and for every y>0 B(x y ) (ii) lim x B(x) =.

9 234 BASSILY, KA TAI, AND WIJSMULLER This means that for fixed k the functions { k (n) belong to the class H. It is now easy to establish the following lemma. Lemma 4.. For each fixed k and every real number z, lim x lim x <z (4.) b k (log log x) k+2 ==8(z), x > { } { k(n)&a k (log log x) k+ <z (4.2) b k (log log x) k+2 ==8(z).?(x) > { } { k( p&)&a k (log log x) k+ Proof. Let H=(=B k (x)) k =(log log x)(=) k (log log x) 2k for some =>0. Since k is fixed, H>C log log x when x is sufficiently large. Therefore, by (2.4) lim x B 2 (x) k { k ( p)<=b k (x) { 2 k( p) p =0 for every =>0. We can now apply the well-known theorem by Kubilius and Shapiro (Theorem 2.2 in [2]) to conclude (4.) while (4.2) follows from the result by Barban, Levin, and Vinogradov (Theorem 2.4 in [2]). This ends the proof of Lemma PROOFS OF THEOREMS AND 2 In this section we assume that k is fixed. In order to compare { k (}) with ( k ( } )) we give the following definition. Definition 5.. A (k+)-tuple of primes (q 0, q,..., q k ) is called a k-chain if q i& q i & for i=, 2,..., k. A general k-chain is denoted by Q k. A k-chain with the property that q k n is denoted by Q k (n) and Q k (n, q 0 ) denotes those k-chains where q 0 is fixed and q k n. Let Q k (n, q 0 ) =>[Q k (n, q 0 )]; then { k (n)= Q q0 k (n) k(n, q 0 ). In order to replace { k (}) by ( k ( } )) in (4.) and (4.2) we must show that { k (n)& ( k (n))=o((log log x) k+2 ) for all but o(x) choices of and, similarly, that { k ( p&)& ( k (p&))=o((log log x) k+2 ) for all but o(li(x)) primes. This will follow from Lemmas 5. and 5.2. Let L(x) = R(x) = ({ k (n)& ( k (n)))=l () +L (2), ({ k (p&)& ( k (p&)))=r () +R (2),

10 PRIME DIVISORS OF. k (n) 235 where L () = R () = q 0 k (n) q 0 <y q 0 k (n) q 0 <y ( Q k (n, q 0 ) &). ( Q k (p&, q 0 ) &). L (2) =L(x)&L (), R (2) =R(x)&R (). Lemma 5.. When y=(log x) 2 L () =O(x(C log log x) k log log log x), R () =O(li(x)(C log log x) k log log log x). Proof. Since L () q 0 k (n) q 0 <y Q k (n, q 0 ) x q 0 <y }}}, q q k q k repeated application of Lemma 2.5 results in L () <<x(c log log x) k log log y=o(x(c log log x) k log log log x). To evaluate R () we observe that R () q 0 k (n) q 0 <y Q k (p&, q 0 ) q 0 <y }}}?(x,q k,)= + +. q q k 2 3 In we consider only q k x 2 which means that <<li(x) q 0 <y }}} <<li(x)(c log log x) k log log y q q k q k =O(li(x)(C log log x) k log log log x).

11 236 BASSILY, KA TAI, AND WIJSMULLER In 2 we consider those q k >x 2 for which { k (q k )(C log log x) k =H k. The number of all k-chains ending at q k is equal to { k (q k ); therefore, << 2 x 2 <q k x { k (q k )?(x, q k,) <<H k?(x)=o(li(x)(c log log x) k ). Finally, using Lemma 2.2 and the estimate (2.4), = 3 q 0 < y }}} q x 2 <q k x { k (q k )>H k?(x, q k,) << x 2 <q k x { k (q k )>H k << \ q k x { k (q k )>H k << \ q k x { k (q k )>H k { k (q k )?(x, q k,) { 2 (q 2 k k) q k + \ 2+ 2 q k (?(x, q k,)) x 2 <q k x { 2 (q 2 k k) li(x)=o q k + log log x)k x \(C log 2 x +. Lemma 5.2. When y=(log x) 2 then x(c log log x) 2k+ L (2) =O \k2 log +, x(c log log x) 2k+ 2 R(2) =O x \k2 log 2 x +. Proof. When q 0 y the number of k-chains starting at q 0 is in most cases. Only those pairs (n, q 0 ) for which Q k (n, q 0 ) 2 will make a contribution to L (2). For integers N2, N&( N 2 ) which means that an upperbound to L (2) is obtained when we consider the number of distinct pairs of k-chains for each pair (n, q 0 ). Let + j (q 0, n) be the number of k-chain pairs Q k (n, q 0 ), Q$ k (n, q 0 )(=(q 0,q$,..., q$ k )) for which q j {q$ j and q l =q$ l for l> j. Then clearly L (2) q 0 y q 0 k (n) (+ (q 0, n)+}}}++ k (q 0,n)) =M +M 2 +}}}+M k, where M j = q0 y + j(q 0, n).

12 PRIME DIVISORS OF. k (n) 237 We begin by evaluating M k. Since q k {q$ k it follows that q k q$ k n. Therefore, M k x }}} q,q 2,..., q k q k q$ k q 0 y q$, q$ 2,..., q$ k q k {q$ k <<x(c log log x) 2 q 0 y }}} q, q 2,..., q k& q$,q$ 2,..., q$ k&. q k& q$ k& To evaluate further we distinguish between two cases q k& =q$ k& and q k& {q$ k&. In the first case we observe that the number of all chains ending at q k& is equal to { k& (q k& ) and, since we are counting pairs of distinct chains ending at q k&, q 0 y }}} q, q 2,..., q k& q 2 k& q$, q$ 2,..., q$ k& q k& =q$ k& << q k& y { 2 k& (q k&) q 2 k& << B2 (x) k&. y When q k& {q$ k& we apply Lemma 2.5 again and repeat the argument. Since all chains start at the same q 0, k M k <<x j= Similarly, when j<k we see that (C log log x) 2j B 2 (x) k& j kx(c log log x)2k+ << y y M j <<x q 0 y }}} }}} q,q 2,..., q j q j+ q$, q$ 2,..., q$ j q j {q$ j q q k k <<x(clog log x) k& j q 0 y }}} q,q 2,..., q j q$, q$ 2,..., q$ j q j {q$ j q j q$ j. Using the same argument as in the evaluation of M k it follows that j M j <<x(c log log x) k& j l= j+ jx(c log log x)k+ <<. y (C log log x) 2l B 2 (x) j&l y Since y=(log x) 2, it follows that L (2) =O(k 2 x(c log log x) 2k+ log 2 x). By the simple observation that R (2) L (2) the proof is complete.

13 238 BASSILY, KA TAI, AND WIJSMULLER It follows from Lemmas 5. and 5.2 that, for fixed k, for almost all integers the difference between { k (n) and ( k (n)) is o((log log x) k+2 ). Also the difference between { k ( p&) and ( k ( p&)) is o((log log x) k+2 ) for almost all primes. This means that { k ( } ) can be replaced by ( k ( } )) in (4.) and (4.2). From Lemma 2. we deduce that 0 kx (. k (n))& ( k (n)) { 0 (n)+ }}} +{ k& (n)k { k& (p) =kx A k& (x). p (n)+}}}+ ( k& (n)) { k& (n) When k is fixed this is o(xb k (x)) and Theorem follows. Similarly, (. k ( p&))& ( k (p&))k { k& (p&) and, since { k& (p&)= q p& { k& (q)= { k (p)=s k (x) which is o(li(x) B k (x)) by Lemma 3. and 3.2, Theorem 2 follows. 6. WHEN k=k(x) DEPENDS UPON x The referee of an earlier draft of this paper suggested that we consider k as a variable depending upon x. In first instance we need to make sure that the error terms in (3.2) and (3.4) are of smaller order than the main term. There is no hidden dependence upon k in the order terms and it follows that k must be o(log log log xlog log log log x). For such k we can no longer use the techniques of Section 4, since our function { k (n) depends upon x. But in this case we can use Theorem 2.5 in [2] and using the lemma of Berry and Esseem (Lemma.48 in [2]) one can show that (4.) and (4.2) still hold. Clearly, (. k (n))&{ k (n) (. k (n))& ( k (n)) + kxa k& (x)+l () +L (2). ( k (n))&{ k (n) When k=o(log log log xlog log log log x) the three quantities at the right are o(xb k (x)) and it follows that (.) holds for such k. A similar argument shows that (.2) also remains valid for such k.

14 PRIME DIVISORS OF. k (n) 239 We thank the referee for the careful reading of the manuscript and for the suggestion that we might strengthen Lemma 2.6, consider k=k(x), and prove Theorems and 2 in a more general form. REFERENCES. E. Bombieri,, On the large sieve, Mathematika 2 (965), P. D. T. A. Elliot, ``Probabilistic Number Theory,'' Springer-Verlag, New York, P. Erdo s, A. Granville, C. Pomerance, and C. Spiro, On the normal behaviour of the iterates of some arithmetic functions, in ``Analytic Number Theory, Proc. Conf. in honor of Paul T. Bateman'' (B. C. Berndt et al., Eds.), pp , Birkha user, Boston, P. Erdo s and C. Pomerance, On the normal number of prime factors of.(n), Rocky Mountain J. 5 (985), H. Halberstam and H. Richert, ``Sieve Methods,'' Academic Press, LondonNew York, I. Ka tai, On the number of prime factors of.(.(n)), Acta. Math. Hungar. 58, Nos. 2 (99), I. Ka tai, Distribution of (_( p+)), Ann. Univ. Sci. Budapest. Sect. Comput. 34 (99), J. Kubilius, ``Probabilistic Methods in the Theory of Numbers,'' Am. Math. Soc. Trans. Math. Monographs, No., Amer. Math. Soc., Providence, RI, M. Ram Murty and V. Kumar Murty, Analogue of the Erdo skac theorem for Fourier coefficients of modular forms, Indian J. Pure Appl. Math. 5 (984), G. Tenenbaum, private communication.