On the average number of divisors of the Euler function

Transcription

1 On the average number of divisors of the Euler function Florian Luca Instituto de Matemáticas Universidad Nacional Autonoma de Méico CP 58089, Morelia, Michoacán, Méico Carl Pomerance Department of Mathematics Dartmouth College Hanover, NH , USA March 3, 2006 Abstract We let ϕ and τ denote the Euler function and the numberof-divisors function, respectively In this paper, we study the average value of τϕn when n ranges in the interval [, ] Mathematics Subject Classification: N37 Key Words: divisors, Euler function The first author was supported in part by UNAM grant PAPIIT IN04505 The second author was supported in part by NSF grant DMS This work was completed while the first author visited Dartmouth College as a Shapiro Fellow

2 Introduction For a positive integer n, let ϕn denote the Euler function of n, and let τn, ωn and Ωn denote the number of divisors of n, the number of prime divisors of n, and the number of prime-power divisors of n, respectively There have been a number of papers that have discussed arithmetic properties of ϕn, many of these inspired by the seminal paper of Erdős [5] from 935 In particular, in [7] see also [6], the normal number of prime factors of ϕn is considered It has been known since Hardy and Ramanujan that the normal value of ωn or Ωn is log log n, and since Erdős and Kac that fn log log n/ log log n has a Gaussian distribution for f = ω or Ω In [7], it is shown that ϕn normally has 2 log log n2 prime factors, counted with or without multiplicity In addition, there is a Gaussian distribution for fϕn log log n2 2 3 log log n 3/2 for f = ω and f = Ω In [2], it is shown that the normal value of Ωϕn ωϕn is log log n log log log log n Note that it is an easy eercise to show that τn is on average log n That is, log n n τn n However, from Hardy and Ramanujan, since 2 ωn τn 2 Ωn, we know that for most numbers n, τn = log n log 2+o, where log 2 = 0693 Thus, τn is on the average somewhat larger than what it is normally Similarly, for most numbers n, τϕn = 2 2 +olog log n2 One might suspect then that on average, τϕn is somewhat larger It comes perhaps as a bit of a shock that the average order of τϕn is considerably larger Our main result is the following: Theorem Let D ϕ := τϕn 2 n

3 Then, the estimate D ϕ = ep c ϕ /2 log + O log log log log log log log holds for large real numbers where c ϕ is a number in the interval and γ is the Euler constant [7 e γ/2, 2 3/2 e γ/2 ], 2 We point out that Theorem above has already been used in the proof of Theorem in [9] to give a sharp error term for a certain sum related to Artin s conjecture on average for composite moduli Recall that the Carmichael function of n, sometimes also referred to as the universal eponent of n and denoted by λn, is the eponent of the multiplicative group of invertible elements modulo n If n = p ν p ν k k is the factorization of n, then λn = lcm[λp ν,, λp ν k k ], where if p ν is a prime power then λp ν = p ν p ecept when p = 2 and ν 3 in which case, λ2 ν = 2 ν 2 It is clear that λn ϕn and that ωλn = ωφn The function Ωϕn/λn = Ωϕn Ωλn was studied in [2] In addition to the result on Ωϕn ωϕn mentioned above, it is shown in [2] that Ωϕn Ωλn log log n log log log log n on a set of n of asymptotic density In the recent paper [], Arnold writes it would be interesting to study eperimentally how are distributed the different divisors of the number ϕn provided by the periods T of the geometric progressions of residues modulo n It is clear that the numbers T range only over the divisors of λn We have the following result Theorem 2 Let D λ := τλn n 3

4 i The estimate D λ = ep c λ /2 log + O log log log log log log log holds for large real numbers where c λ is a number in the interval shown at 2 ii With D ϕ = ma y D ϕ y, the estimate holds as D λ = od ϕ Concerning part ii of Theorem 2, we suspect that even the sharper estimate D λ = od ϕ holds as, but we were unable to prove this statement We mention that in [3], in the course of investigating sparse RSA eponents, it was shown that τϕn log n Ωn=2 see [3], page 347 In particular, the average value of the function τϕn over those positive integers n which are the product of two primes is bounded above by a constant multiple of log 2 /log log Our methods can also be applied to study the average number of divisors of values of other multiplicative functions as well For eample, assume that f : N Z is a multiplicative function with the property that there eists a polynomial P k Z[X] of degree k with P 0 0 such that fp k = P k p holds for all prime numbers p and all positive integers k For any positive integer n we shall write τfn for the number of divisors of the nonnegative integer fn, with the convention that τ0 = In this case, our methods show that there eist two positive constants α and β, depending only on the polynomial P, such that the estimate τfn = ep c n /2 log + O log log 4 log log log log log

5 holds for large values of with some number c [α, β] In particular, the same estimate as holds if we replace the function ϕn by the function σn Indeed, the lower bound follows eactly as in the proof of Theorem, while for the upper bound one only needs to slightly adapt our argument We close this section by pointing out that it could be very interesting to study the average value of the number of divisors of fn for some other integer valued arithmetic functions f We mention three instances Let a > be a fied positive integer and let fn be the multiplicative order of a modulo n if a is coprime to n and 0 otherwise We recall that the functions ωfn and Ωfn were studied by Murty and Saidak in [] It would be interesting to study the average order of τfn in comparison with that of τλn Let E be an elliptic curve defined over Q Let fn be the multiplicative function which on prime powers p k equals p k + a p k, the number of points of E defined over the finite field F p k with p k elements, including the point at infinity Let fn be the Ramanujan τ function which is the coefficient of q n in the formal identity 24 q k = + fnq n k= We believe that it should be interesting to study the average number of divisors of fn for these functions fn and other multiplicative functions that arise from modular forms Perhaps the methods from this paper dealing with the easy case of ϕn will be of help A relevant paper here is by Murty and Murty [0] in which, building on work of Serre [2, 3], the function ωfn is analyzed, where fn is the Ramanujan τ function Throughout this paper, we use c, c 2, to denote computable positive constants and to denote a positive real number We also use the Landau symbols O and o, the Vinogradov symbols and, and the equal-order-ofmagnitude symbol with their usual meanings For a positive integer k we use log k for the recursively defined function log := ma{log, } and log k := ma{loglog k, } where log denotes the natural logarithm function When k = we simply write log as log and we therefore understand that log always We write p and q for prime numbers For 5 n

6 two positive integers a and b we write [a, b] for the least common multiple of a and b Acknowledgements The authors thank the anonymous referee for a careful reading of the manuscript and for suggestions that improved the quality of this paper 2 Some Lemmas Throughout this section, A, A, A 2, A 3, B and C are positive numbers We write z := z for a function of the real positive variable which tends to infinity with in a way which will be made more precise below We write P z := p z p The results in this section hold probably in larger ranges than the ones indicated, but the present formulations are enough for our purposes For any integer n 2 we write pn and P n for the smallest and largest prime factor of n, respectively, and we let p = +, P = Lemma 3 Assume that z log / log 2 i For any A > 0 there eists B := BA such that if QP z <, we then have log B E z := µr π; n, π ϕn log A 3 r P z n Q r n The constant implied in depends at most on A ii Let A, A, A 2 > 0 be arbitrary positive numbers Assume that u is a positive integer with pu > z, u < log A and τu < A 2 There eists B := BA, A, A 2 such that if QP z < log B, then E u,z := r P z µr n Q r n π; [u, n], π ϕ[u, n] The constant implied in depends at most on A, A, A 2 log A 4 6

7 Proof Note that µr r P z n Q r n ψ; n, ϕn = n P z n 2 Q γ n δ n2 ψ; n n 2,, 5 ϕn n 2 where γ n := µn if P n z and it is zero otherwise, and δ n2 := for all n 2 Q Similarly, µr ψ; [u, n], ϕ[u, n] r P z n Q r n = n P z n 2 Q γ n δ n 2 ψ; n n 2,, 6 ϕn n 2 where γ n := γ n and δ n 2 := 0 if u n 2, and it is the cardinality of the set {d Q [d, u] = n 2 } otherwise Note that if n 2 Q, then δ n 2 = τu is a constant ie, does not depend on n 2 provided that δ n 2 is nonzero The same argument as the one used in the proof of Theorem 9 in [4] leads to the conclusion that both 5 and 6 are of order of magnitude at most /log A provided that B is suitably large in terms of A and of A, A and A 2, respectively Now 3 and 4 follow from 5 and 6 by partial summation and using the fact that these sums are of order of magnitude at most /log A From now on until the end of the paper we use c for the constant e γ, where γ is the Euler constant Lemma 4 Let A > 0 and < z log A We have and L z := n pn>z M z := n pn>z n = c log log z + O ϕn = c log log z + O log log 2 z The constants implied by the above O s depend only on A 7, 7 log log 2 8 z

8 Proof Write K z := n pn>z By Brun s Sieve see Theorems 22 on page 68 and 25 on page 82 in [8], we have that K z = c log z + O log 2 if z < / log 2, 9 z and K z log z if z 0 We shall now assume that z < / log3 2 Using partial summation, we have Clearly, L z = dk z t t = K z + K z = O log z K z t dt t 2 by estimate 9 We break the integral at / log 2 By estimate 0, we get / log 2 K z t dt / log 2 dt t 2 log z t log log z log 2 2 For the second range we use estimate 9 to get K z t dt = c dt + O / log 2 t 2 log z log z / log 2 t = c log + O + O 3 log z log z log 2 Collecting together all estimates 3 we get L z = c log log log z + O log 2 z + log log z log 2 + log z and it is easy to see that the above error is bounded above as in 7 when z log A, as in the hypothesis of the lemma 8,

9 For 8, note that M z = ϕn = n pn>z = d pd>z d> pd>z µ 2 d dϕd n pn>z m /d pm>z µ 2 d n ϕd d n m = d pd>z µ 2 d dϕd L z/d When d =, µ 2 d/dϕd =, while when d >, since pd > z, it follows that µ 2 d dϕd dϕd z, d>z where the last estimate above is due to Landau Thus, M z = L z + <d z pd = c log log z + O µ 2 d dϕd L z/d = L z + O log log 2 z, which completes the proof of the lemma For, z > 0, let D z = {n : pn > z} Lz z and let τ z m be the number of divisors of m in D z m Lemma 5 Let A > 0 and z A log log 4 2 We then have R z := τ z p = c log z + O p log 2 z 4 and S z := p τ z p p log log = c log z + O log 2, 5 z where the constants implied in O above depend only on A 9

10 Proof Let y be any positive real number Our plan is to estimate R z y, so proving 4, and then use partial summation to prove 5 Note that R z y = τ z p τp 2 πy; d, y, 6 p y p y d y where the last estimate follows from the Brun Titchmarsh inequality Corollary in [4] gives a more precise estimate We shall use this estimate when y is relatively small In general, R z y = τ z p = = πy; d, p y p y d D zy d p d D zy Assume now that y > e z log2 z We write B for a constant to be determined later If y is large, we then split the sum appearing in R z y at y Q := P z log B y Then, R z y = πy; d, + πy; d, := R + R 2 7 d D zq Q<d y pd>z Note that if d > Q and p y is a prime with p mod d, then p = +du with u < y/q = P z log B y Thus, R 2 πy; u, u P z log B y By the Brun Titchmarsh inequality, we get R 2 πy u P z log B y ϕu πy logp z log B y y log 2 y log y + yz log y y log 2 z, 8 where the last inequality above holds because y > e z log2 z We now deal with R We claim that R = πy d D zq ϕd + O y log 2 y 9 0

11 holds if B is suitably chosen Thus, Indeed, note that by the principle of inclusion and eclusion, we have R = d Q pd>z πy; d, = r P z µr n Q r n R = E z y + r P z µr n Q r n πy; n, πy ϕn, where E z y has been defined in Lemma 3 By Lemma 3, the estimate R = µr πy y ϕn + O y = πy log C y ϕd + O log C y r P z n Q d D zq r n holds with any value of C > 0 provided that B is chosen to be sufficiently large with respect to C We set C := 2, and we obtain 9 Since y Q = P z log B y > y/2 > ep 2 z log2 z, it follows that z Q/log 2 Q, and we are therefore entitled to apply Lemma 4 and conclude that πy log Q πy log Q y R = c + O log z log 2 + O z log 2 y y y = c log z + O log 2 20 z Combining 7 20, we get that y R z y = c log z + O y log 2 z 2 holds when y > e z log2 z, which in particular proves estimate 4 To arrive at 5, we now simply use partial summation to get that dr z t S z = = R zt t= R z t + dt t t t= t 2 e z log 2 z R z t R z t = dt + dt + O t 2 e z log2 z t 2

12 The first integral above is, by 6, e z log 2 z R z t t 2 dt e z log 2 z t dt z log2 z log log 2 z, 22 while the second integral above is, by 2, R z t dt = c e z log2 z t 2 log z e z log2 z t dt + O log 2 z log log = c log z c z log z + O log 2 z = c log log z + O and 5 now follows from 22 and 23 log log 2 z e z log2 z t dt, 23 Lemma 6 i Let A and z be as in Lemma 5 and u be any positive integer with pu > z Then S u,z := p p mod u τ z p p τu u S z log 24 ii Let A > 0, 0 < A 2 < /2, u < log A and log A 2 < z Assume that pu > z Then and R u,z := p p mod u τ z p = c τu u log z + O u log 2 z log log S u,z = τu u S z + O 26 log z The implied constants depend at most on A and A, A 2, respectively Proof To see inequality 24, we replace the prime summand p with an integer summand n, so that S u,z n n mod u τ z n n 2 = n n mod u n d D z d n

13 Thus, S u,z d D z log n mod [u,d] n d D z [u, d] n = log τu u d D z d D z = τu u L z log τu u S z log, [u, d] where in the above inequalities we used Lemmas 4 and 5 d m /[u,d] For inequality 26, let us first notice that under the conditions ii, we have that Ωu ; hence, τu, and also that uϕd ϕud = + O 27 z holds uniformly in such positive integers u and all positive integers d The proof of 26 now closely follows the method of proof of 5 That is, let be large, assume that z is fied, and for y write R u,z y := τ z p = πy; [u, d], p y p mod u d D zy log Let w := ep Note that for large the inequality z < log y log 2 log 2 y 4 holds whenever y > w For y w, we use the trivial inequality m R u,z y y log y 28 u Assume now that y > w Since log y > log /3 holds for large, and u < log A, we get that u < log 3A y We write B for a constant to be y determined later and we split the sum appearing in R u,z y at Q := P z log B y Thus, R u,z y = πy; [u, d], + πy; [u, d], := R + R 2 d D zq 3 Q<d y pd>z

14 It is easy to see that R 2 πy; [u, d], d P z log B y Thus, by the Brun Titchmarsh inequality, R 2 πy d P z log B y y log 2 y u log y + yz u log y ϕ[u, d] πyτu u logp z log 3A +B y y u log 2 z, 29 where we used τu together with 27 We now deal with R We claim, as in the proof of Lemma 5, that y R = πy ϕ[u, d] + O u log 2 y d D zq 30 holds if B is suitably chosen Indeed, note that since u and P z are coprime, by the principle of inclusion and eclusion, we have Thus, R = d Q pd>z R = E u,z y+ r P z µr n Q r n πy; [u, d], = r P z µr n Q r n πy ϕ[u, n] = E u,zy+πy πy; [u, n], d D zq ϕ[u, d], 3 where E u,z y is the sum appearing in Lemma 3 Estimate 30 now follows from 4 Since Q u > y P z u log B > y P z log > 3A +B y/2 > ep 2 z log4 z, 4

15 it follows that z Q/u log 4 Q/u, and we are therefore entitled to apply Lemmas 4 and 5 together with estimate 27 and conclude that y R = πy ϕ[u, d] + O u log 2 y d D zq τu = πy ϕud + O πy y ϕud + O u log 2 y d D zq/u Q/u<d Q pd>z = πy τu πy u M zq/u + O uz M zq πy y +O u M zq M z Q/u + O u log 2 y = c πy τu logq/u πy log u πy log Q + O + O u log z u log z u log 2 z τu y y = c u log z + O u log 2 z 32 Combining 28 32, we get that R u,z y = c τu u y log z + O y u log 2 z 33 holds when y > w, which proves estimate 25 We now use partial summation to get that S u,z = = w dr u,z t t R u,z t dt + t 2 The first integral above is, by 28, = R u,zt t w R u,z t t 2 t= + t= dt + O R u,z t dt t 2 u log z w R u,z t t 2 dt u w log t t dt log2 w u log u log 2 2 log u log 2 z 34 5

16 Finally, the second integral above is, by 33, R u,z t τu dt = c w t 2 u log z w t dt + O u log 2 z w t dt τu log = c log log w + O u log z u log 2 z τu log log = c u log z + O u log 2 z τu = c u S z + O, log z which completes the proof of Lemma 6 log Lemma 7 Let z = z := log 6 2 Let I := z, z5 ] Let Q be the set of all prime numbers p with z < p such that p is not divisible by the square of any prime q > z, and p has at most 7 prime factors in the interval I Then for large we have S z := p Q τ z p p > 07S z Proof Let Q be the set of those primes p such that q 2 p for some q > z Fi q Assume first that q > log A, where A is a constant to be determined later Then, by 24, and therefore q>log A S q 2,z log S q 2 z, S q 2,z log S z Choosing A =, we see that q>log q>log A q 2 S q 2,z S z log 2 S z log A log 2 6

17 Assume now that q z, log ] By 26, it follows that S q 2,z S z /q 2, and therefore z<q log S q 2,z S z q>z q S z 2 z log z S z log 2 In particular, we have q>z log S q 2,z = O log We now let B be a positive integer to be fied later, and assume that u is a squarefree number having ωu = B, and such that all its prime factors are in the interval I Let U B be the set of such numbers u Since B is fied, we have u < z 5B < log 5B/2, and therefore, by 26, we have S u,z = 2B u S z + O log 2 Summing up over all possible values of u U B, we get S u,z = S z 2 B + O u log u U B u U 2 B Clearly, u u U B B! Hence, = log 5B B! p I u U B S u,z B p + O = B log B! 2 z 5 log 2 z + O log z log 2 2 log 5B S z + O B! log 2 Since 2 log 5 8 /8! < 0286, we have for B = 8 and large that u U B S u,z < 029S z 36 7

18 Thus, with 35 and 36, and using p z τ zp /p = p z /p log 2 z, S z = p Q τ z p p S z q>z S q 2,z u U B S u,z p z for large, which completes the proof of Lemma 7 3 The Proof of Theorem We shall analyze the epression 3 The upper bound T := n τϕn n τ z p p > 07S z For every positive integer n we write βn := p n p Then n can be written as n = βnm where all prime factors of m are among the prime factors of βn Moreover, ϕn = mϕβn, and therefore τϕn τmτϕβn Thus, where T k m /k µk 0 τmτϕk mk = k µk 0 τϕk k m /k τm m U log 2, 37 U := n µn 0 τϕn n We now let z = z be as in Lemma 7 For every positive integer n we write τ z n for the number of divisors of the largest divisor of n composed only of primes p z Clearly, τn = τ z nτ z n If n and pα n, then α + < 2 log This shows that log τ zn 2 log πz < ep 0 log

19 Using also the fact that τ z ab τ z aτ z b holds for all positive integers a and b, together with 37 and 38, we get that the inequality log T V z ep O log 5 2 holds, where V z := n µn 0 p n τ z p p To find an upper bound on the last epression, we use Rankin s method Let s = s < be a small positive real number depending on to be determined later, and note that V z s n µn 0 s p n s p n τ z p p + τ zp p +s = s n ep We now find s in such a way as to minimize f s := s log + p p n µn 0 τ z p p +s s log + p τ z p p +s For this, recall that from the proof of Lemma 5, we have p τ z p p +s = 2 = + s dr z t t +s 2 = R zt t= t +s t=2 R z t t 2+s dt + O τ z p p +s R z t + + s dt 2 t 2+s 39 s log z We shall later choose s := c /2 log log z In order to compute the above integral 39, we split it at 0 := e /s log2 z In the first smaller range, we use the fact that R z t t and that t s to get 0 2 R z t t 2+s dt 0 2 t dt log 0 = 9 s log 2 z

20 Note that 0 e z log2 z for sufficiently large Thus, from the estimate of R z t from Lemma 5, we have R z t dt = c dt 0 t 2+s log z 0 t + O dt +s log 2 z 0 t +s c = s log z s 0 s + O s log 2 z = c s log z + O s log 2 z where we used the fact that s 0 = ep/log2 z = + O/log 2 z Thus, f s = s log + c + s + O s log z s log 2 z With our choice for s, we have f s = 2c /2 Since z = log /log 6 2, we get f s = 2 3/2 c /2, /2 log log /2 + O log z log 3/2 z /2 log + O log 2 Thus, we have obtained the upper bound log T ep 2 3/2 e γ/2 log 2 /2 + O log3 log 2 Since D ϕ T, we have the upper bound in Theorem 32 The lower bound We write c 2 for a constant to be computed later and we set /2 log v := c 2 2 log 2 20 log3 40 log 2

21 We write y := ep log log c 2 /2 2 We now write z := zy, where the function z is the one appearing in Lemma 7 We write Q := Qy for the set of primes defined in Lemma 7 Recall that Qy is the set of primes p y such that p is not divisible by the square of any prime q > z and p has at most 7 distinct prime factors in z, z 5 ] Consider squarefree numbers n having ωn = v and such that all their prime factors are in Q Let N be the set of those numbers It is clear that if n N then n /y 2 Let V N,z := n N τ z ϕn 4 n For a number n N, we write τ n := p n τ zp and we look at the sum W N := τ n n n N By the binomial formula, and Stirling s formula, it follows that W N v τ z p v e τ z p v! p v v p p Q p Q A simple calculation based on Lemmas 5 and 7 shows that 07e S zy v e v p Q τ z p p e S zy v and that S z y v = c 3 + O log3, log 2 2

22 where c 3 := 4c We now observe that v! c 2 2 p Q W N + = W N + O v τ z p p v 2! v! p Q p Q v 2 τ z p τ z p 2 p 2 p 2 p Q v τ z p τ z p 2 p p 2 Since p Q implies that p > z, and since for large the inequality τ z p τp < p /4 holds for all p > z, we get τ z p 2 p 2 p>z p Q Hence, the above argument shows that W N = τ z p v! p p Q v c 4 + O log3 log 2 p Q p 3/2 z /2 log z v + O v, z /2 log z where c 4 := 07c 3 We now select the subset M of N formed only by those n such that there is no prime number q > z 5 such that q p and q p 2 holds for two distinct primes p and p 2 dividing n Note that this is equivalent to the fact that ϕn is not a multiple of a square of a prime q > z 5 To understand the sum W M restricted only to those n M, let us fi a prime number q Then, summing up τ n/n only over those n such that q 2 ϕn for the fied prime q > z 5, we get W q,n := n N q 2 ϕn τ n n W N S q,z y 2 v 2! p Q 22 v 2 τ z p S q,zy 2 p

23 Assume first that q > log A y, where A is a constant to be determined later In this case, by Lemmas 6 and 5, and therefore S q,z y S zy log y q W q,n W N log2 y q log z, Summing up inequalities 42 for all q log A y, we get q log A y W q,n W N log 4 y log 2 z q>log A y log 4 y q 2 log 2 z 42 q 2 W N log y A 4 log 2 2 y W N log y, provided that we choose A := 5 When q < log 5 y, then the same argument based again on Lemma 6, shows that log 2 y W q,n W N q 2 log 2 z, and therefore z 5 <q<log 5 y W q,n W N log 2 y log 2 z q>z 5 q 2 W N log 2 y z 5 log 3 z W N log 27 2 y log /2 y This shows that W M := n M τ n n W N q>z 5 W q,n = W N + o v c 4 + O Notice now that if n M then τ z ϕn τ z 5ϕn = p n Thus, with 4 and 43, we get v log3 43 log 2 τ z 5p p n V N,z 2 7v W M v c 5 + O 23 τz p 2 7 v log3, log 2 2 7v τ n 44

24 where c 5 := 2 7 c 4 So, we see that log V N,z ep c 6 log 2 /2 + O log3, 45 log 2 holds for large with 07 4e γ c 6 := c 2 log c To see the lower bound in Theorem, we look at integers np, where n M and p is prime with y < p /n Each such integer np arises in a unique way, and the number of primes p corresponding to a particular n is π/n πy π/n /n log Further, τϕnp > τ 2 zϕn Thus, D ϕ log V N,z 47 So, letting c 2 = 28/2 7 /2 e γ/2 > 7 e γ/2, we have c 6 = c 2, and from 45 and 47, we have the lower bound in Theorem for all large 4 The proof of Theorem 2 Part i follows immediately from the proof of Theorem Indeed, λn ϕn, therefore τλn τϕn holds for all positive integers n In particular, D λ D ϕ For the lower bound, it suffices to note that if M is the set of integers constructed in the proof of the lower bound for D ϕ, then ϕn is not divisible by the square of any prime p > z 5 In particular, the inequality τ z λn = τ n/2 7v also holds compare with 44 Thus, the lower bound on D λ follows from the proof of the lower bound for D ϕ To see ii, we put κ = 0 log 2 and w = log log 2 2, and let E be the set of n such that either 2 κ n or there eists a prime p n with p mod 2 κ, and E 2 be the set of n with ωn w 24

25 Since τφab τφaτφb, we have n E n E τλn n n E τϕn n τ2κ τϕm 2 κ m + m /2 κ p p mod 2 κ τp p m /p τϕm m We majorize the inner sums with T, so that τλn κ + + S n 2 κ 2 κ, T κ log S 2 κ T log2 log 2 T log 0 log 2 T log 2, 48 where in the above estimates we used Lemmas 5 and 6 to estimate S 2 κ, and S, respectively, and the fact that 0 log 2 > 4 Furthermore, by the multinomial formula and the Stirling formula, n E 2 τλn n n E 2 k w k w k w τϕn n k! p α k! p k w τϕp α p α τϕp p n ωn=k k + 2 p 2 α k! S + O k k w τϕn n τϕp α p α k es + O k Let c 7 be the constant implied by the last O Since S log, the k 25

26 k es + c 7 function k is increasing for k w once is large Thus, k w τλn es + c 7 w n w n E 2 es + c 7 = ep w log + log w w log T = ep 2 + o = o log 2 log 2, 49 where the last estimate above follows from estimate 45 and T V N,z Finally, if we set E 3 for the set of all positive integers n not in E E 2, we then notice that if n E 2, then 2 α ϕn, where α ωn w, and 2 β λn, where β < κ Hence, Thus, τϕn τλn n E 3 α + β + log log 3 2 τλn log3 2 log D ϕ, 50 while estimates 48, 49 and partial summation show that Therefore log 2 n E E 2 τλn log 2 n E E 2 dtd ϕ t T = D ϕ + t Dϕ dt + Dϕ log t n E E 2 Clearly, summing up estimates 50 and 5 we get τλn n D ϕ t dt t τλn D ϕ log 5 D λ log3 2 log D ϕ, and the proof of Theorem 2 is complete 26

27 References [] V I Arnold, Number-theoretical turbulence in Fermat-Euler arithmetics and large Young diagrams geometry statistics, J Math Fluid Mech , suppl, S4 S50 [2] W D Banks, F Luca and I E Shparlinski, Arithmetic properties of ϕn/λn and the structure of the multiplicative group modulo n, Comment Math Helvetici , 22 [3] W D Banks and I E Shparlinski, On the number of sparse RSA eponents, J Number Theory 95 no , [4] E Bombieri, J B Friedlander and H Iwaniec, Primes in arithmetic progressions to large moduli, Acta Math 56 no , [5] P Erdős, On the normal number of prime factors of p and some other related problems concerning Euler s ϕ function, Quart J Math Oford Ser 6 935, [6] P Erdős, A Granville, C Pomerance and C Spiro, On the normal behaviour of the iterates of some arithmetic functions, in: Analytic Number Theory, Proc of a Conference in Honour of PT Bateman, Birkhäuser; Boston, Inc 990, [7] P Erdős and C Pomerance, On the normal number of prime factors of ϕn, Rocky Mtn J of Math 5 985, [8] H Halberstam and H-E Richert, Sieve methods, Academic Press, 974 [9] S Li, An improvement of Artin s conjecture on average for composite moduli, Mathematika, to appear [0] M R Murty and V K Murty, Prime divisors of Fourier coefficients of modular forms, Duke Mathematical Journal 5 984, [] M R Murty and F Saidak, Non-abelian generalizations of the Erdős- Kac theorem, Canad J Math , [2] J-P Serre, Divisibilité des certaines fonctions arithmétiques, L Ens Math ,

28 [3] J-P Serre, Quelques aplications du théorème de densité de Chebotarev, Inst Hautes Études Sci Publ Math 54 98,