Smooth Values of Shifted Primes in Arithmetic Progressions

Smooth Values of Shifted Primes in Arithmetic Progressions William D. Banks Department of Mathematics, University of Missouri Columbia, MO 65211 USA bbanks@math.missouri.edu Asma Harcharras Department of Mathematics, University of Missouri Columbia, MO 65211 USA harchars@math.missouri.edu Igor E. Shparlinski Department of Computing, Macuarie University Sydney, NSW 2109, Australia igor@ics.m.edu.au Abstract We study the problem of bounding the number of primes p x in an arithmetic progression for which the largest prime factor of p h does not exceed y. 1 Introduction Recall that an integer n 1 is said to be y-smooth if it is not divisible by any prime p exceeding y. Smooth numbers have played an important role in many Corresponding author 1

number theoretic and cryptographic investigations, and there is an extensive body of literature on the subject, originating with the work of Dickman [5] and de Bruijn [3]. For an interesting account of smooth numbers, we refer the reader to the survey article by Granville [11]; see also the references contained therein. As usual, we denote by Ψx, y) the counting function for smooth numbers: Ψx, y) = #{n : 1 n x and n is y-smooth}. It is well-known that the asymptotic relation Ψx, y) ρu) x holds in a very wide range within the xy-plane, where u = log x)/) and ρu) is the Dickman-de Bruijn function which is defined by and ρu) = 1 ρu) = 1, 0 u 1, u 1 ρv 1) dv, u > 1. v For an account of the basic analytic properties of ρu), we refer the reader to the book by Tenenbaum [19]. In this paper, we are interested in finding upper bounds for the number of primes p x that lie in a fixed arithmetic progression and such that p h is y-smooth, where h 0 is a fixed integer. To describe our results, let us introduce some notation that is used throughout the seuel. As usual, we denote by πx) the prime counting function: πx) = #{prime p x}. Next, following [17], we define πx, y) = #{prime p x : p 1 is y-smooth}. More generally, for any integer h 0, let π h x, y) = #{prime p x : p h is y-smooth}. Finally, for any integers, a with 1 and gcda, ) = 1, we put π h x, y;, a) = #{prime p x : p h is y-smooth and p a mod )}. 2

Since one might expect that the set {p 1 : prime p x} contains roughly the same proportion of y-smooth integers as the set of all positive integers n x, it is reasonable to conjecture following Erdős [6]; see also [16]) that the relation πx, y) ρu) πx) holds for all x and y in a fairly wide range. More generally, for fixed h 0 one might also conjecture see, for example, [14]) that the relation π h x, y) ρu) πx) also holds in a wide range. Indeed, for all primes p > h the uantity p h is relatively prime to h, thus it is reasonable to expect that the set {p h : prime p x} contains roughly the same proportion of y-smooth integers as the set of all positive integers n x coprime to h, and for a fixed value of h the latter proportion is easily seen to be ρu) using a standard sieve to detect coprimality. Finally, arguing that the set of primes p x such that p h is y-smooth is likely to be evenly distributed over all admissible arithmetic progressions modulo that is, those that represent infinitely many primes), it may be true that the relation π h x, y;, a) ρu) πx) 1) ϕ) holds in a wide range. At the present time, however, all of these conjectures appear to be out of reach. Over the years, substantial progress has been made on the problem of finding upper and lower bounds for πx, y); see [1, 2, 4, 6, 11, 14, 15, 16]. In particular, highly nontrivial upper bounds for πx, y) have been found by Fouvry and others in the case where y > x 1/2, while for smaller values of y, the bound πx, y) = O u ρu) πx)) has been recently obtained by Pomerance and Shparlinski [17] in the range ) exp log x log log x y x. 2) In the shorter range exp log x) 2/3+ε) y x, 3) 3

the slightly stronger estimate πx, y) ρu)πx) follows from Theorem 4 of Fouvry and Tenenbaum [8]. In this paper, we show how the methods of [17] combined with results of Granville [10] see also [9]) and Fouvry and Tenenbaum [7] on smooth integers in arithmetic progressions can be used to prove that the estimate ) u ρu) ch) π h x, y;, a) = O πx) 4) ϕ) holds uniformly with respect to each of the involved parameters) for all x, y and in a wide range, where ch) = p 1 p 2. p h p 2 Moreover, we obtain an explicit value though somewhat modest) for the implied constant in 4). For a precise statement of this result, see Theorem 2 below. We remark that the conjecture 1) suggests that our upper bound 4) is probably not tight as it contains the extra factors u and ch); it remains an interesting open uestion as to whether these factors can be removed from 4). Our result immediately implies that rather sparse arithmetic progressions contain a positive proportion of primes for which p 1 has a large prime divisor; see the discussion in Section 5. Our result also has an interesting cryptographic conseuence. It is well known that primes p for which p 1 is smooth are not suitable for most cryptographic applications derived from Diffie-Hellman or RSA schemes. The results of [17] show that in fact such primes are very rare. However, primes p such that p 1 is not smooth are occasionally chosen to satisfy some additional conditions; for example, for the applications described in [20], what is needed is a good supply of non-smooth primes lying in a certain arithmetic progression. The results of this paper imply, in a uantitative form, that almost all primes p in any arithmetic progression with a modulus of moderate size are such that p 1 is not smooth. Throughout the paper, the implied constants in symbols O, and may occasionally depend, where obvious, on the small parameter ε > 0 but 4

are absolute otherwise. We recall that the expressions A B and B A are each euivalent to the statement that A = OB). Throughout, the letters p and l always denote prime numbers, while n and always denote positive integers. Acknowledgements. We wish to extend a special note of thanks to Glyn Harman, whose detailed comments on the manuscript helped to improve the exposition and the uality of our results. We also thank Roger Baker and Carl Pomerance for several enlightening discussions. The first two authors would like to thank Macuarie University for its hospitality during the preparation of this paper. Work supported in part, for the first author, by NSF grant DMS-0070628, and for the third author, by ARC grant DP0211459. 2 Preliminaries Here we collect some preliminary estimates to be used in the seuel. For any integer n 2, let P + n) denote the largest prime divisor of n, and put P + 1) = 1. As in Section 1, we define Ψx, y) = #{n x : P + n) y}. Throughout the seuel, the parameter u is defined, as usual, to be the ratio u = log x)/) whenever x and y are given. The following result of Hildebrand [13] concerns the asymptotic nature of the function Ψx, y); see also Corollary 9.3 in Chapter III.5 of [19]. Lemma 1. For every ε > 0, the estimate )) logu + 1) Ψx, y) = ρu) x 1 + O holds uniformly provided that exp log log x) 5/3+ε) y x. For any integers and a with gcda, ) = 1, let Ψx, y;, a) = #{n x : P + n) y and n a mod )}, and put Ψ x, y) = #{n x : P + n) y and gcdn, ) = 1}. 5

We need the following result of Granville [10] about smooth numbers lying in a fixed arithmetic progression. Lemma 2. For any ε > 0, the estimate Ψx, y;, a) = 1 ϕ) Ψ x, y) 1 + O log u c + 1 )) holds uniformly provided that gcda, ) = 1 and 1+ε y x, for some constant c > 0 that depends only on ε. Here, as usual, ϕ denotes the Euler function. We also need the following result of Fouvry and Tenenbaum [7] about smooth numbers relatively prime to a fixed modulus. Lemma 3. For any ε > 0, the estimate Ψ x, y) = ϕ) )) log logy) log log x Ψx, y) 1 + O holds uniformly provided that x x 0 ε), exp log log x) 5/3+ε) y x, and log log + 2) ) 1 ε. logu + 1) We also need the following two lemmas concerning the Dickman-de Bruijn ρ-function. Lemma 4. For any u 0 and 0 δ u with δ logu+1) 0, the following estimate holds: ρu δ) = ρu) 1 + O δ logu + 1))). Proof. In Lemma 1 of [13], we find the estimate ρ u) ρu) logu log2 u), u e 4. In particular, ρ u) ρu) logu + 1), u 0. 5) 6

If 0 δ u, then for some v in the interval [u δ, u], we have that is, ρu δ) ρu) = δ ρ v) δρv) logv + 1) δρu δ) logu + 1); ρu) = ρu δ) 1 + Oδ logu + 1))). Taking into account that δ logu+1) 0, we obtain the desired estimate. Lemma 5. For all u 1, we have u 1 ρt) dt u + 1 + 1/u)ρu). Proof. Using the well known identity see, for example, Lemma 1 of [13]) ρu) = 1 u u u 1 ρt) dt, u 1, it follows that ρu) ρu 1)/u; by induction, we obtain the estimate ρu + j) Thus, for all u 1, we have ρt) dt = u + j) u 1 j=0 This completes the proof. ρu) u + j)u + j 1)...u + 1), j 1. u+j u+j 1 uρu) + ρu) + ρu) ρt) dt = j=2 u + j)ρu + j) j=0 1 = u + 1 + 1/u)ρu). u + 1) j 1 Finally, we recall the following result from sieve theory; see Theorem 3.12 from [12]. Lemma 6. Let m, h, and b be integers such that mh 0, gcdm, h) = 1, 2 mh, gcd, b) = 1, and 1 log Y ) A, where A > 0 is a fixed constant. Then, as Y, the number of primes l Y such that l b mod ) and ml + h is prime is at most 8 ) 1 )) p 1 Y log log Y 1 p 1) 2 p 2 ϕ) log 2 1 + O A, Y log Y p>2 2<p mh uniformly in the parameters m, h, and b. 7

3 An Asymptotic Formula Our principal tool is the following theorem, which is an extension of Lemma 1 from [17] and which we believe to be of independent interest. While for the purposes of this paper we reuire only an upper bound for the sum considered below, we remark that Theorem 1 in fact provides an asymptotic formula in certain wide ranges of the involved parameters. Theorem 1. Let ε > 0 be fixed. For any real numbers x and y satisfying exp log log x) 5/3+ε) y x, any integer 1 y 1 ε, and any integer a with gcda, ) = 1, we have where m x, P + m) y m a mod ) m ϕm) = ζ 2)ζ 3) ρu) x 1 + O E x, y))), ζ 6) E x, y) = log log log log x + + y log x log log x logu + 1), x and ζ s) is the partial zeta-function defined for Rs) > 1 by ζ s) = p 1 p s ) 1. Proof. For any integer d with gcdd, ) = 1, denote by d the uniue integer such that 1 d and dd 1 mod ). Then m x, P + m) y m a mod ) m ϕm) = = = m x, P + m) y m a mod ) d x, P + d) y gcdd,)=1 d x, P + d) y gcdd,)=1 8 d m µd) 2 ϕd) µd) 2 ϕd) m x/d, P + m) y m ad mod ) 1 µd) 2 ϕd) Ψx/d, y;, ad ).

For the moment, suppose that d x/y. Since 1+ε y x/d, Lemma 2 provides the uniform estimate Ψx/d, y;, ad ) = Ψ x/d, y) log 1 + O ϕ) u c d + 1 )) for some constant c > 0 depending only on ε, where u d = logx/d))/). If 2, then since u d 1, we have Ψx/d, y;, ad ) = Ψ )) x/d, y) log 1 + O. 6) ϕ) Clearly, this estimate also holds when = 1. Next, we want to apply Lemma 3 to estimate Ψ x/d, y). To do this, we need to check that the necessary conditions on x/d and y are met. First, observe that exp log logx/d)) 5/3+ε) exp log log x) 5/3+ε) y x/d. Also, x/d y x 0 ε) if x is sufficiently large. Finally, if x and y are large enough, then u d + 1 = logx/d) log x + 1 + 1 log x, log log x) 5/3+ε logu d + 1) log log x ) log 1/5/3+ε) y)2/5, and therefore ) 1 ε log log + 2) log logy 1 ε + 2) ) 21 ε)/5. logu d + 1) Applying Lemma 3, we obtain the uniform estimate Ψ x/d, y) = ϕ) )) log logy) log logx/d) Ψx/d, y) 1 + O. Because we therefore see that Ψ x/d, y) = ϕ) log logy) log 2 ε = Olog ), Ψx/d, y) 1 + O 9 log log log x )). 7)

Combining the estimates 6) and 7), we now derive that Ψx/d, y;, ad ) = 1 + O Ψx/d, y) log log log log x + )) 8) provided that d x/y. For any d > x/y, we also have Ψx/d, y;, ad ) = n x/d n ad mod ) 1 = x + O1). 9) d Now put z = min{, x/y}. Using 8) and 9), it follows that m x, P + m) y m a mod ) m ϕm) = = Σ 1 + Σ 2 ) d x, P + d) y gcdd,)=1 1 + O µd) 2 ϕd) Ψx/d, y;, ad ) log log log log x + )) + OΣ 3 + Σ 4 ), where Σ 1 = Σ 2 = Σ 3 = Σ 4 = d z, P + d) y gcdd,)=1 µd) 2 ϕd) z<d x/y, P + d) y gcdd,)=1 x/y<d x, P + d) y gcdd,)=1 x/y<d x, P + d) y gcdd,)=1 Ψx/d, y), µd) 2 ϕd) µd) 2 ϕd) µd) 2 ϕd). Ψx/d, y), x d, 10

First, let us estimate Σ 1. For all d z, we have by Lemma 1: Ψx/d, y) = ρu d ) x )) logud + 1) 1 + O d ) )) log x log d x log log x = ρ 1 + O. d Also, by Lemma 4, we have ) log x log d ρ since if δ = log d)/), then = ρu) 1 + O log log log x )), 0 δ logu + 1) log log log x, and the last term tends to 0 as x when x and y lie in the specified range. Thus, we derive that Σ 1 = ρu) x )) log log log x µd) 2 1 + O dϕd). d z, P + d) y gcdd,)=1 Finally, by the well known ineuality for the Euler function d z, P + d) y gcdd,)=1 ϕd) d, d 3, 10) log log d see, for example, Theorem 5.1 from Chapter 1 of [18]), we have µd) 2 µd) 2 = dϕd) dϕd) + O log log d d 2 d 1 gcdd,)=1 = ζ 2)ζ 3) ζ 6) d> log log + O ). ) Noting that 1 < ζ 2)ζ 3) ζ 6) < 2, 1, 11) 11

it now follows that Σ 1 = ζ 2)ζ 3) ζ 6) ρu) x 1 + O log log log x )). To estimate Σ 2, we may assume that z = < x/y, since Σ 2 = 0 otherwise. Under this assumption, put j 0 = log z = log and j 1 = logx/y) = log x. Then, using Lemma 1 again, we derive that and therefore Σ 2 = 1 1 x Σ 2 x <d x/y, P + d) y gcdd,)=1 <d x/y j 0 j j 1 ρ µd) 2 ϕd) Ψx/d, y) µd) 2 log x log d ϕd) ρ u j + 1 ) ) x d µd)2 dϕd), e j <d ej+1 log j ρ u j + 1 ). 12) e j j 0 j j 1 Here we have used 10) and the fact that ρu) is a decreasing function of u. Next, we observe that the function ft) = logt 1) e t 1) ρ u t ) satisfies the estimate ft + 1) ft) = 1 e 1 + O )) log log x 13) for all t in the range t log x 1. Indeed, by Lemma 4, we have for all t log x 1: ρ u t + 1 ) = ρ u t ) )) log log x 1 + O, 12

since if δ = 1/, then 0 δ log u t ) + 1 log log x 0. The estimate 13) follows immediately. This shows that ft) is a function of exponential decay provided that x is sufficiently large; hence from 12) we now derive that Σ 2 x j 0 e j 0 ρ u j ) 0 + 1. Since j 0 = log + O1), we have ) j 0 e j 0 log = O, while another application of Lemma 4 gives ρ u j ) )) 0 + 1 log log log x = ρu) 1 + O. Using 11), it follows that Σ 2 ρu) x log 1 + O )) log log log x Σ 1 log. Finally, we turn to the estimates for Σ 3 and Σ 4. Put j 2 = log x. Using 10), we derive that Σ 3 = x x x/y<d x, P + d) y gcdd,)=1 j 1 j j 2 e j <d e j+1 P + d) y µd) 2 dϕd) x log log d d 2 x/y<d x P + d) y x log log d d 2 log j e 2j j 1 j j 2 Ψe j+1, y). By Lemma 1, Ψe j+1, y) = e j+1 ρ ) j + 1 1 + O log log x )), 13

and therefore Σ 3 x log j ρ e j j 1 j j 2 ) j + 1 x log j 1 ρ e j 1 ) j1 + 1 y log log x ρu 1). Since ρ u) = ρu 1)/u for all u 1, the estimate 5) implies Conseuently, Σ 3 y log log x For Σ 4, we have the estimate Σ 4 = By Lemma 1, Σ 4 x/y<d x, P + d) y gcdd,)=1 log log x x ρu 1) u logu + 1) ρu), u 1. y log x log log x logu + 1) u logu + 1) ρu) Σ 1. x µd) 2 ϕd) ρu) x 1 + O x/y<d x P + d) y log log d d )) logu + 1) log log x x Ψx, y). log log x Σ 1. x Combining our estimates for Σ j, 1 j 4, and using the fact that < y, the result follows. Corollary 1. Let ε > 0 be fixed. For any real numbers x and y satisfying exp log log x) 5/3+ε) y x, any integer with log = o), and any integer a with gcda, ) = 1, we have m ϕm) ζ 2)ζ 3) ρu) x. ζ 6) m x, P + m) y m a mod ) Proof. Examining the structure of the error term in Theorem 1, it follows that E x, y) = o1) provided that exp log log x) 5/3+ε) y 14 x log log x) 1+ε

and log = o), and we obtain the reuired asymptotic formula. For larger values of y, one can argue directly as follows: m ϕm) = m ϕm) + m ϕm). m x, P + m) y m a mod ) m x m a mod ) For the first summation, it is easy to check that m x m a mod ) m x, P + m)>y m a mod ) m ϕm) = ζ 2)ζ 3) ρu) x ζ 6) while for the second summation, we have m ϕm) r log log x gcdr,)=1 x ϕ) log x 1 + O y<p x p ar mod ) r log log x m x, P + m)>y m a mod ) log log log x r ϕr) pr ϕpr) log x )), x log log + 2) log log x log log log log x log x ) ρu) x = o. Here we have used the Brun-Titchmarsh theorem for the second ineuality; see Theorem 2.2 of [12]. This completes the proof. 4 Main Results As in Section 1, let us define for all n 1, cn) = l 1 l 2. l n l 2 Then we have the following trivial estimate: cn) l n l 2 l l 1 l>2 l 1) 2 ll 2) = 15 n C 2 gcdn, 2) ϕn), 14)

where C 2 is the twin primes constant given by C 2 = ) 1 1 = 0.6601618158.... p 1) 2 p>2 Theorem 2. Let ε > 0 and A > 0 be fixed. Then for any real numbers x and y satisfying ) exp log x log log x) 1+ε y x with x x 0 ε, A), the estimate ) ζ 2)ζ 3) u + 1 + 1/u)ρu)ch) π h x, y;, a) 8δ h + o1) πx) ζ 6) holds provided that ) A, gcda, ) = 1, and h 0, where δ h = 1 if h is even, and δ h = 1/2 if h is odd. Proof. In what follows, l always denotes a prime number. Let π h,l x;, a) = #{p x : P + p h) = l and p a mod )}. Let z = exp log log x) 2 ), and suppose that z Y x. Then, assuming that x is sufficiently large, we have π h x, Y ;, a) π h x, Y/e;, a) = π h,l x;, a) = Y/e<l Y = 1 b gcdb,)=1 1 b gcdb,)=1 Y/e<l Y m x h)/l, P + m) l ml+h prime ml+h a mod ) Y/e<l Y l b mod ) m xe/y, P + m) Y mb+h a mod ) 1 m x h)/l, P + m) l ml+h prime mb+h a mod ) Y/e<l Y l b mod ) ml+h prime 1. 1 16

By Lemma 6, we have the estimate Y/e<l Y l b mod ) ml+h prime 1 8C 2 + o1)) cmh) Y ϕ) log 2 Y. Note that the summation on the left hand side is actually 0 for sufficiently large x) if gcdm, h) > 1 or if both m and h are odd. Using the trivial ineuality cmh) ch)cm) and the estimate 14), it therefore follows that π h x, Y ;, a) π h x, Y/e;, a) 8δ h + o1)) By Corollary 1, we have ch) Y ϕ) log 2 Y π h x, Y ;, a) π h x, Y/e;, a) ch) Y 8δ h η + o1)) ϕ) log 2 Y where 8δ h η + o1)) ch) x log 2 Y ρ 1 b gcdb,)=1 1 b gcdb,)=1 log x log Y 1 η = ζ 2)ζ 3). ζ 6) m xe/y, P + m) Y mb+h a mod ) m ϕm). log x log Y + 1 ρ log Y ), Now let j 0 = logy/z). Using the above estimate, we have π h x, y;, a) π h x, z;, a) + j 0 j=0 1 + o1))ψx, z;, a) + 8δ h η + o1)) ch) x ) x Y πh x, y/e j ;, a) π h x, y/e j+1 ;, a) ) j 0 j=0 ) 1 log x j) ρ 2 j 1. 17

Arguing as on page 341 of [17], we have the estimate j 0 j=0 ) 1 log x j) ρ 2 j 1 ρu 1) log 2 y If y lies in the stated range, then as x, ρu 1) log 2 y = o u ρu) log x ), + 1 ρt) dt. 15) log x u 1 while the integral on the right hand side of 15) can be estimated using Lemma 5. Finally, using 8) with d = 1 together with Lemma 1, we have Ψx, z;, a) = The result follows. Ψx, z) 1 + o1)) = x ) log x ρ 1 + o1)). log log x) 2 We remark that the bound 4) stated in the introduction follows immediately from Theorem 2 using the trivial estimate ch) c)ch) together with 14). 5 Concluding Remarks We note that the range of Theorem 2 is slightly shorter than that of [17] given by 2). However, one can easily extend Theorem 2 to the same range with o1) replaced by O1); in this case, we lose the explicit form of the statement. The integral in Lemma 5 can be calculated precisely for certain values of u. For example, u 1 ρt) dt = { e γ + 1 u if 1 u 2, e γ + 3 2u + u 1) logu 1) if 2 u 3, where γ is the Euler-Mascheroni constant; similar expressions can be obtained whenever u is reasonably small and can be used in place of u+1+1/u)ρu) 18

in the statement of Theorem 2 to obtain better estimates. In this way, we find that 4 ζ2)ζ3) C 2 ζ6) u 1 ρt) dt < 1, u > u 0 = 3.3558619258..., hence it follows that for any fixed A > 0, we have lim inf x # {p x : p a mod ), P + p 1) x 0.295 } min > 1 1 log x) A πx)/ϕ)) 16, gcda,)=1 where P + n) is the largest prime divisor of n. We also remark that the pair 0.295, 1/16) can be replaced with 0.270, 1/2) or 0.257, 2/3), for instance. For = 1, the current record with the exponent 0.677 instead of 0.295 has been established in [2] though one loses the positivity in the density of primes), and it seems likely that by appropriately modifying the techniues of that paper, this stronger result might also be obtained for primes in arithmetic progressions. This has never been worked out explicitly, however, and doing so would necessarily entail many technical and tedious calculations; thus Theorem 2 provides a reasonably painless shortcut to albeit weaker) results of the same general type. In fact, the range of is uite generous and coincides with the range for which unconditional results have been obtained on the asymptotic formula for primes in arithmetic progressions. It is easy to see that any improvement of our principal tool, Lemma 6, will immediately lead to an improvement of Theorem 2 both with respect to the constant 8 and the range of ). One can also try to prove an analogue of Theorem 1 for the sum cm). m x, P + m) y m a mod ) Using 14), one sees that this sum is bounded by ζ 2)ζ 3) C 2 ζ 6) ρu) x 1 + O E x, y))), and any improvement in this estimate would lead to a corresponding improvement of Theorem 2. 19

In principle, it should be possible to improve the bound in Theorem 2 by about a factor of u) using the approach of [8], but as in the case of = 1) only in a narrower range of y; compare 2) and 3). As we have already mentioned any further progress to close the gap between 1) and Theorem 2 would be of great interest. Finally, we remark that by using similar techniues, it should be possible to obtain analogues of our results for primes p x in an arithmetic progression such that kp + h is y-smooth. References [1] W. R. Alford, A. Granville and C. Pomerance, There are infinitely many Carmichael numbers, Annals Math. 140 1994), 703-722. [2] R. C. Baker and G. Harman, Shifted primes without large prime factors, Acta Arith. 83 1998), 331 361. [3] N. de Bruijn, On the number of positive integers x and free of prime factors > y, Nederl. Acad. Wetensch. Proc. Ser. A 54 1951), 50 60. [4] C. Dartyge, G. Martin and G. Tenenbaum, Polynomial values free of large prime factors, Periodica Math. Hungar. 43 2001), 111 119. [5] K. Dickman, On the freuency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astr. Fys. 22 1930), 1 14. [6] P. Erdős, On the normal number of prime factors of p 1 and some other related problems concerning Euler s φ-function, Quart. J. Math. Oxford Ser.) 6 1935), 205 213. [7] É. Fouvry and G. Tenenbaum, Entiers sans grand facteur premier en progressions arithmetiues, Proc. London Math. Soc. 3) 63 1991), 449 494. [8] É. Fouvry and G. Tenenbaum, Répartition statistiue des entiers sans grand facteur premier dans les progressions arithmétiues, Proc. London Math. Soc. 3) 72 1996), 481 514. [9] A. Granville, Integers, without large prime factors, in arithmetic progressions. I., Acta Math. 170 1993), 255 273. 20

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