Short sums of certain arithmetic functions - IECL

Acta Math. 180 (1998), 119–144. c �1998 by Institut Mittag Le✏er.

Short sums of certain arithmetic functions Mohan Nair & G´erald Tenenbaum

1. Introduction Let M denote the class of all non-negative multiplicative functions g with the property that (⌫ 2 N, p prime), (i) 9 A : g(p⌫ ) 6 A⌫ (ii) 8" > 0 9 B = B(") > 0 : g(n) 6 Bn" (n 2 N).

In 1980, Shiu [7] obtained a general upper bound for short sums of functions g 2 M : Let ↵, β 2]0, 1[, and let x, y satisfy x > y > x↵ . Then for positive integers a, q with (a, q) = 1 we have n X g(p) o X y g(n) ⌧ exp '(q) log x p p6x, p - q

x 0, X

(6)

x x"1 −"2 = x"2 with q 6 x"3 . Hence there is an h, 1 6 h 6 r, such that q ⌫ |Rh (n) with q ⌫ > q e/r > x3"3 . Let ⌫(q) denote the least integer such that q ⌫(q) > x3"3 . Then ⌫(q) > 3 and q ⌫(q)−1 6 x3"3 so q ⌫(q) 6 q 4"3 6 y. Since q ⌫(q) |Q⇤ (n) we may hence write X y%⇤ (q ⌫(q) ) . S2 6 2 max F (|Q1 (n)|, . . . , |Qk (n)|) n62x q ⌫(q) q6x"3 With the bound F (|Q1 (n)|, . . . , |Qk (n)|) 6 G(|Q1 (n)|, . . . , |Qk (n)|) 6 B|Q(n)|"δ/3 , we deduce that y (33) S2 ⌧ kQk"δ/3 x"δ/3 3"3 x"3 ⌧ yx−"/3 , x

taking account of the assumption that kQk 6 x1/δ . Now we note that the right-hand side of (18) is � y/(log x)g since v(1; F ; %) = F (1, . . . , 1) = 1 and %(p) 6 g for all p. Hence the estimate (33) for S2 is also of the required order of magnitude. Estimation of S3 and S4 . We begin by an estimate which is common to S3 and S4 . For all n in C3 [ C4 , we have bn = |Q(n)|/an 6 (g + 1)kQk(2x)g /x"2 6 xg+1/δ . Since

P + (an )⌦(bn ) < P − (bn )⌦(bn ) 6 bn ,

we deduce from (30) that with

e 1n , . . . , brn ) ⌧ xE(an ) , G(b

� E(a) := min g", s/ log P + (a) ,

s := (g + 1/δ) log A.

Therefore, using (12), we may write X 00 Fe(m1 , . . . , mr )xE(m ) (34) S3 + S4 ⌧ γ

"1 r m1 1 ···mγ r 6x

X

1.

xP + (m1 ···mr )

Employing the sieve as in S1 to bound the inner sum, we arrive at Y ⇣ X %(p) ⌘ (35) S3 + S4 ⌧ y v(n; F, σ )xE(n) 1− , p r + " " x

2 /D

w. e 1 , . . . , nr ) 6 Bn"δ/3 ⌧ x"/3 , and %h (nh ) 6 g !(nh ) when Since Fe(n1 , . . . , nr ) 6 G(n (nh , D) = 1 we may write, using the trivial bound of 1 for the product over p in (35), g"

S3 ⌧ yx

X

n1 ,...,nr r

x"2 /D ( h=1 nγhh )1/g � x"3 /g . We now choose r

w := 2AgD 4(s+1)/"2 .

(36)

This implies that the last sum over n is ⌧ (log x)w , and in turn shows that S3 is of the required order of magnitude. It remains to estimate S4 . We consider the sub-sum of (35) corresponding to P + (n) > w, which we expand by writing n = q ⌫ m with P + (m) < q. We observe that v(n; F, σ ) 6 (Ag/q)⌫ v(m; F, σ ) by (12) and (13), and hence we get Y⇣ %(p) ⌘ 1− T3 p

S3 ⌧ y

p6x

with T3 :=

X

xs/ log q

w 4g . We obtain T3 ⌧ ⌧

X

−1/ log q

x

w c0 kP kδ . It remains to confirm that these conditions hold. Now (x − a)2g

2

"β 1−2g 2 "β

q

6 y β/2 y (1−β)(1−2g

2

"β)

6 y 1−β/2 6 y,

so (i) holds. Next, using kP k 6 q g kQk and x > c1 kQk2δ , we have that x > c1 kP k2δ q −2gδ , hence xq > c1 kP k2δ and thus x2 > c1 kP k2δ . So the choice c1 = c20 suffices to confirm (iii). Finally, for (ii), it is easily checked that if x − a < y 6 x then we have the trivial estimate ⇣y x − a ⌘ X F (|P1 (m)|, . . . , |Pk (m)|) ⌧ − + 1 kP k"βδ/6 q q 2(x−a)/q 1 and we are in a position to apply Corollary 2 which implies that X

x z0 with arbitrary large z0 = z0 (, ", δ, g). Indeed, when 2 2/2g " 6 z < z0 , the left-hand side of (21) is trivially ⌧ (log x)O(1) , which is of smaller order of magnitude than the right-hand side. Let β := / log z and let χ(n) denote the completely multiplicative function defined by χ(p) = pβ if p 6 z, χ(p) = 1 otherwise, and put F1 (n1 , . . . , nk ) := F (n1 , . . . , nk )χ(nk ). We have χ(n) = nβ when P + (n) 6 z. Since |Qk (n)| � x for x < n 6 x + y, the sum on the right-hand side of (21) is ⌧ x−β

X

x 0 by the relation 2" = (1 + ⌘)/ log 4. Applying Corollary 4 with t = 1 + ", we obtain S2 ⌧ y(log x)−σ+o(1) with σ = 1 + ⌘" − β(1 + ") = 2{2" log(2" ) − 2" + 1} > 0. Thus S2 = o(y) and, by (44), HQ (x, y) � y/(log x)⌘ . � By (43), this implies Px,y > y exp (log x)1−⌘1 for any ⌘1 > ⌘ and large x. The required result follows since ⌘, and hence ⌘1 , may be taken arbitrarily close from log 4 − 1.

9. Proof of Theorem 3 We may plainly assume that c0 > 1 and F (1, . . . , 1) = 1. We first observe that it is sufficient to bound the sub-sum of (22) where the variable p is further restricted by the condition p - Q(0). Indeed, the complementary contribution may be trivially bounded above by !(|Q(0)|)B sup |Q(p)|"δ/3 . x x0 (δ), p 6 B(log x)(2g xg+1/δ )"δ/3 ⌧B (log x)x"(g+1)/3 ⌧ y. This is (with a lot to spare) of smaller order of magnitude than the right-hand side of (22). Let χ(n) = 1 if P − (n) > x, and χ(n) = 0 otherwise. We set F0 (n1 , . . . , nk+1 ) = F (n1 , . . . , nk )χ(nk+1 ). It is readily checked that our hypothesis on F implies that F0 2 Mk+1 (A, B, "δ/3). Since Q(0) 6= 0, none of the Qj (X) is equal to X, so b Q(X) := X

k Y

j=1

Qj (X)

Short sums of certain arithmetic functions

21

has degree g+1 and has r+1 irreducible factors Rj (X) (1 6 j 6 r+1), with Rr+1 (X) = X. b = kQk. We set %b := % , %b := (%1 , . . . , %r+1 ) and note that %r+1 = 1. Of course kQk b Q Now we make the observation that if the variable of summation n in Theorem 1 is restricted to values coprime to a fixed integer, say q, then the implicit constant in the ⌧ sign only depends on Dq := D/(D, q). This may be easily checked by taking into account the extra condition (n, q) = 1 in the sieve arguments employed for the upper bounds of S1 , S3 , S4 in section 4, so we omit the details. b ⇤ . Then D b ⇤ = a2g⇤ Q⇤ (0)2 D⇤ , hence D b ⇤ denote the discriminant of (Q) b q |D when Let D q = Q(0). It follows that Theorem 1 (suitably modified as indicated above) yields the bound X X � � � � F |Q1 (p)|, . . . , |Qk (p)| 6 F0 |Q1 (n)|, . . . , |Qk (n)| x