Character sums with exponential functions over smooth numbers

Transcription

1 Indag. Mathem., N.S., 17 (2), June 19, 2006 Character sums with exponential functions over smooth numbers by William D. Banks a, John B. Friedlander b, Moubariz Z. Garaev c and Igor E. Shparlinski d a Dept. of Mathematics, University of Missouri, Columbia, MO 65211, USA b Dept. of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada c Inst. de Matemáticas, Universidad Nacional Autónoma de México, C.P , Morelia, Michoacán, México d Dept. of Computing, Macquarie University, Sydney, NSW 2109, Australia Communicated by Prof. R. Tijdeman at the meeting of June 20, 2005 ABSTRACT We give nontrivial bounds in various ranges for exponential sums of the form exp(2πiaϑ n /m) n S(x,y) and n S t (x,y) exp(2πiaϑ n /m), where m 2, ϑ is an element of order t in the multiplicative group Z m, gcd(a, m) = 1, S(x, y) is the set of y-smooth integers n x, ands t (x, y) is the subset of S(x, y) consisting of integers that are coprime to t. We obtain sharper bounds in the special case that m = q is a prime number. 1. INTRODUCTION For an integer m 1, letz m denote the ring of integers modulo m, andletz m be the group of units in Z m ; recall that Z m can be identified with the set of congruence classes a(mod m) with gcd(a, m) = 1. Letϑ be a fixed element in Z m of multiplicative order t 1. There is a rich history of study which involves character sums with the exponential function ϑ n (and the distribution of its values) as the integer n varies over the interval [1,x]; see [16 20] for a variety of results in this direction and MSC: 11L07, 11L20 s: bbanks@math.missouri.edu (W.D. Banks), frdlndr@math.toronto.edu (J.B. Friedlander), garaev@matmor.unam.mx (M.Z. Garaev), igor@ics.mq.edu.au (I.E. Shparlinski). 157

2 their numerous applications. More recently, several results have been obtained for character sums of the form (1) e m (aϑ n ), a Z m, n S where e m (z) = exp(2πiz/m) for all z R,andS is a subset of the integers in [1,x] defined by certain arithmetical conditions, including: S = the set of primes or the set of squarefree integers (cf. [1]); S = the set of squares or higher powers (cf. [7,8]); S = a cyclic subgroup of Z t (cf. [7 9]); S = the set of products of elements from two arbitrary (sufficiently large) subsets of Z t (cf. [10]); S = the set of squarefull or, more generally, k-full integers (cf. [5,21]); S = the set of integers having a prescribed sum of g-ary digits in a fixed base g 2 (cf. [2,11]). We remark that, for many sets S (especially those that are quite thin), the associated character sums (1) are much harder to control than sums which correspond to the full set of integers in the interval [1,x]. In this paper, we initiate the study of character sums (1) in the special case that S is a collection of smooth numbers in [1,x]. More precisely, recall that a positive integer n is said to be y-smooth provided that P(n) y, wherep(n) is the largest prime factor of n (with the usual convention that P(1) = 1). Let S(x, y) denote the set of all y-smooth positive integers n x, andlets t (x, y) be the subset of integers n S(x, y) with gcd(n, t) = 1. Here, we consider the problem of bounding the character sums T a (m,x,y)= n S(x,y) e m (aϑ n ) and T a (m,x,y)= n S t (x,y) e m (aϑ n ), where a is a fixed element of Z m (note that there is no loss in generality in assuming that a and m are coprime). We obtain nontrivial bounds for these sums for a wide range in the xy-plane, provided that t is sufficiently large. All of our bounds are uniform over all integers a Z m and over all integers ϑ Z m with the same multiplicative order t. 2. PREPARATION In what follows, the letters p,q,r always signify prime numbers, x,y,z real numbers, and n a positive integer. Our methods require bounds for exponential sums with ϑ p as p varies over the set of prime numbers; more explicitly, we need bounds for the sums e m (aϑ p ), a Z m, 158 p x

3 where ϑ is a fixed element of Z m. For a composite m, we apply Theorem 3.1 of [1] directly, which implies the following bound: Lemma 1. Let m be a positive integer, ϑ an integer coprime to m, and t the multiplicative order of ϑ modulo m.then max e m (aϑ p ) ( xt 11/32 m 5/16 + x 5/6 t 5/48 m 7/24) x o(1). a Z m p x In the special case that m = q is prime, the results of [12] (in particular, the bound stated in Lemma 5 below) yield the following improvement to Theorem 3.2 of [1]: Lemma 2. Let q be a prime number, ϑ an integer not divisible by q, and t the multiplicative order of ϑ modulo q.then max e q (aϑ p ) ( xt 1/4 q 1/8 + x 5/6 t 2/9 q 1/6) x o(1). a Z q p x The bound of Lemma 2 is nontrivial when t q 1/2+δ for some fixed δ>0. If ϑ has a smaller multiplicative order (in fact, as small as q δ ), we can apply the following estimate contained in [3]: Lemma 3. Let q be a prime number, ϑ an integer not divisible by q, and t the multiplicative order of ϑ modulo q. For every δ > 0, there exists η > 0 such that for all t q δ and x t 2+δ, the following bound holds: max e q (aϑ p ) a Z q p x x log x q η, where the implied constant depends only on δ. We also need the following upper bound for weighted double sums, which is a simplified version of Lemma 2.5 of [1]: Lemma 4. Let m be a positive integer, ϑ an integer coprime to m, and t the multiplicative order of ϑ modulo m. LetK,L,X,Y be real numbers with X, Y > 0. Then, for any sequence of complex numbers (γ l ) supported on the interval [L,L + Y ] and bounded by γ l 1, and for any integer a coprime to m, we have K<k K+X L<l L+Y γ l e m (aϑ kl ) X 1/2 Y 1/2 (X/t + 1) 1/2 (Y/t + 1) 1/2 t 21/32 m 5/16+o(1). 159

4 In the special case that m = q is a prime number, Lemma 2.7 of [1] provides a sharper estimate than that of Lemma 4 above. However, an even stronger bound is implied by Corollary 3 of [12]. Lemma 5. Let q be a prime number, ϑ an integer not divisible by q, and t the multiplicative order of ϑ modulo q. LetK,L,X,Y be real numbers with X, Y > 0. Then, for any sequence of complex numbers (γ l ) supported on the interval [L,L + Y ] and bounded by γ l 1, and for any integer a such that q a, we have K<k K+X L<l L+Y γ l e q (aϑ kl ) X 1/2 Y 1/2 (X/t + 1) 1/2 (Y/t + 1) 1/2 t 3/4 q 1/8+o(1). As before, Lemma 5 is nontrivial only when t q 1/2+δ for some fixed δ>0. If the multiplicative order of ϑ is smaller than this, we can use the following result from [3]: Lemma 6. Let q be a prime number, ϑ an integer not divisible by q, and t the multiplicative order of ϑ modulo q.letk, L, X, Y be real numbers with X, Y > 0. Then, for every δ>0, there exists η>0 such that for any sequence of complex numbers (γ l ) supported on the interval [L,L + Y ] and bounded by γ l 1, and for any integer a such that q a, we have K<k K+X L<l L+Y γ l e q (aϑ kl ) X 1/2 Y 1/2 (X/t + 1) 1/2 (Y/t + 1) 1/2 q η, provided that t q δ, where the implied constant depends only on δ. The following result, which is Lemma 10.1 of [22], helps us to relate the double sums of Lemmas 4 and 5 to the sums over smooth numbers: Lemma 7. Suppose that 2 y z<n x and n S(x, y). Then there exists a unique triple (p,u,v) of integers with the properties: (i) n = uv; (ii) u S(x/v, p); (iii) z<v zp; (iv) p v; (v) If r v,thenp r y. 3. MAIN RESULTS Theorem 8. Let m be a positive integer, ϑ an integer coprime to m, and t the multiplicative order of ϑ modulo m. Then the bound 160

5 max { Ta (m,x,y), T a (m,x,y) } ( t 11/32 m 5/16 + y 1/6 t 5/48 m 7/24) x 1+o(1) holds for all integers a coprime to m. Proof. Let a be fixed. We have T a (m,x,y)= n x e m (aϑ n ) n x P(n)>y e m (aϑ n ). For the first sum, the well-known bounds for exponential sums with ϑ n over the interval [1,x] (see [16 20]) lead to the estimate (2) e m (aϑ n ) (x/t + 1)m 1/2 log m; n x for example, see Lemma 2.1 of [1]. We now concentrate on the second sum. For each integer n included in the sum, we have a unique representation n = kp with a prime p>yand a positive integer k such that P(k) p. Hence, n x P(n)>y e m (aϑ n ) = p>y k x/p P(k) p e m (aϑ kp ). Taking L k = max{y,p(k) 1}, it follows that (3) p>y k x/p P(k) p e m (aϑ kp ) = k<x/y L k <p x/k e m (aϑ kp ). Using Lemma 1 to estimate the inner sum over p, and noting that the order of ϑ k modulo p is t/d k,whered k = gcd(k, t), we obtain the bound k<x/y L k <p x/k e m (aϑ kp ) ( (x/k)(t/dk ) 11/32 m 5/16 + (x/k) 5/6 (t/d k ) 5/48 m 7/24) x o(1) k<x/y x 1+o(1) t 11/32 m 5/16 k<x/y + x 5/6+o(1) t 5/48 m 7/24 k<x/y d 11/32 k k 1 d 5/48 k k 5/6. Collecting together those integers k for which d k is a fixed divisor d of t, wesee that 161

6 k<x/y d 11/32 k k 1 d t d 11/32 τ(t)log x, s x/yd(ds) 1 = d t d 21/32 s x/yd s 1 where τ(t) is the number of divisors of t. Wealsohave k<x/y d 5/48 k k 5/6 k 5/6 x 1/6 y 1/6 τ(t). k<x/y Using the well-known bound τ(t)= t o(1) (see, for example, Theorem 317 of [15]) and remarking that the bound is trivial for t>x, we obtain that e m (aϑ kp ) ( t 11/32 m 5/16 + y 1/6 t 5/48 m 7/24) x 1+o(1). k<x/y L k <p x/k Therefore, taking into account the estimate (2), we derive the bound T a (m,x,y) (x/t + 1)m 1/2 log m + ( t 11/32 m 5/16 + y 1/6 t 5/48 m 7/24) x 1+o(1). Since the first term never dominates, we obtain the desired estimate for the sums T a (m,x,y). For the sums Ta (m,x,y), we apply the inclusion exclusion principle to detect the coprimality condition gcd(n, t) = 1. Thus, T a (m,x,y)= d t µ(d) = d t d S(x,y) n S(x,y) d n µ(d) e m (aϑ n ) n S(x/d,y) e m (aϑ dn ), where, as usual, µ( ) denotes the Möbius function. Hence, for gcd(a, m) = 1, we have Ta (m,x,y) (x/d) 1+o(1)( (t/d) 11/32 m 5/16 + y 1/6 (t/d) 5/48 m 7/24) d t x 1+o(1) t 11/32 m 5/16 d t d 21/32 + x 1+o(1) y 1/6 t 5/48 m 7/24 d t d 53/48. Estimating both sums over d t as τ(t)= t o(1), we complete the proof. In the case that m = q is a prime number, Lemma 2 can be used instead of Lemma 1 in the preceding proof to obtain the following bound: 162

7 Theorem 9. Let q be a prime number, ϑ an integer not divisible by q, and t the multiplicative order of ϑ modulo q.then max { Ta (q,x,y), T a (q,x,y) } x 1+o(1)( t 1/4 q 1/8 + y 1/6 t 2/9 q 1/6) holds for all integers a such that q a. In turn, Lemma 3 combined with some elementary arguments produces: Theorem 10. Let q be a prime number, ϑ an integer not divisible by q, and t the multiplicative order of ϑ modulo q. Then for every δ>0, there exists η>0 such that for all t q δ and y t 2+δ, the bound max { Ta (q,x,y), T a (q,x,y) } η log x xq log y holds for all integers a not divisible by q, where the implied constant depends only on δ. Proof. We follow the same lines as in the proof of Theorem 8 except that, instead of (2), we apply the bound (4) e q (aϑ n ) (x/t + 1)tq η 0 n x from [4] (which holds with some η 0 > 0 depending only on δ) and instead of Lemma 1 we apply Lemma 3 to the inner sum in (3) if d k q δ/2.finally,wealso estimate the contribution from the larger values of d k trivially as 1 k x/y d k q δ/2 x k log(x/k) d t d q δ/2 1 k x/y d k τ(t) q δ/2 x log x log y, x k log y log x log y d t d q δ/2 where, as before, τ(t) denotes the number of divisors of t. Using the bound τ(t)= t o(1), we conclude the proof. In certain ranges (for example, when y is small relative to t and m), a different approach based on the Vaughan identity (see [22]) leads to sharper estimates. Theorem 11. Let m be a positive integer, ϑ an integer coprime to m, and t the multiplicative order of ϑ modulo m. Then the bound max { T a (m,x,y), T a (m,x,y) } xy 1/2 t 11/32 m 5/16+o(1) + t 2 log 5 y holds for all integers a coprime to m provided that t y/log 2 y. x d 163

8 Proof. Here it is more convenient to estimate the sums Ta (m,x,y) and then derive the desired bound for the sums T a (m,x,y). Let z be a fixed real number such that 2 y z x. It follows from Lemma 7 that T a (m,x,y)= n S t (x,y) n>z e m (aϑ n ) + O(z) = p y p t U a (p,x,y,z)+ O(z), where U a (p,x,y,z)= e m (aϑ uv ), v Q(p,y,z) u S t (x/v,p) and Q(p,y,z)={v: z<v zp, p v, and if r v, then p r y}. Writing v = pw, we obtain the bound: U a (p,x,y,z) z/p<w z u S t (x/wp,p) e m (aϑ pwu ). For any real number in the range p/z <1/2,let { } z M(p, z) = 2p (1 + )j :0 j N, where (5) N = log(2p) log(1 + ) 1 log p. Note that 1 A for all A M(p, z); therefore, U a (p,x,y,z) A<w A(1+ ) u S t (x/wp,p) A<w A(1+ ) = = ( A<w A(1+ ) ( u S t (x/ap,p) u S t (x/ap,p) W a (p,x,z,a)+ O( 2 xn/p), e m (aϑ pwu ) ) e m (aϑ pwu ) + x/ap ) e m (aϑ pwu ) + O( 2 x/p) where W a (p,x,z,a)= A<w A(1+ ) u S t (x/ap,p) e m (aϑ pwu ). 164

9 Since p t, it follows that the multiplicative order of ϑ p in Z m is also t. Applying Lemma 4 with ϑ p instead of ϑ, wederivethat Consequently, where W a (p,x,z,a) ( ) x 1/2 ( A ( A) 1/2 Ap t ( x = p ) 1/2 ( x + 1 ) 1/2 ( x pt 2 + A + x t Apt + 1 ) 1/2 Apt + 1 t 21/32 m 5/16+o(1) ) 1/2 t 21/32 m 5/16+o(1). U a (p,x,y,z) ( )t 21/32 m 5/16+o(1) + 1 = x pt ( 2 ) x 1/2 2 = pt 1 = xn pt ( x 2 ) 1/2 3 = p 2 t ( ) x 1/2 4 = p x log p, pt A 1/2 ( ) xz 1/2, pt ( x A 1/2 2 ) 1/2, pzt ( ) x 1/2 ( ) x 1/2 1 = N log p. p p Here, we have used (5) together with the trivial estimates: A 1/2 1 z 1/2, A 1/2 1 (p/z) 1/2. x log p, p Taking z = (x/ ) 1/2 (to balance 2 and 3 )and = x 1 t 2 log 4 y (to balance 3 and 4 ), we see that ( x log p U a (p,x,y,z) + pt ) x p 1/2 t log y t 21/32 m 5/16+o(1) + t2 log 4 y log p. p We can assume that t<(x/2) 1/2 / log 2 y since otherwise the bound is trivial. Therefore <1/2 and the required lower bound p/z follows from our assumption that t y/log 2 y. Summing the preceding estimate over all p y, p t, we obtain: 165

10 T a (m,x,y) = p y p t ( x log y t U a (p,x,y,z)+ O(z) ) + xy1/2 t 21/32 m 5/16+o(1) + t 2 log 5 y + t log y xy1/2 log y t 11/32 m 5/16+o(1) + t 2 log 5 y. x t log 2 y Noting that the bound is trivial when y m, we can assume that log y = m o(1) and thus remove it (for simplicity) without incurring any loss in our estimate. Next, we observe that T a (m,x,y)= d t n S(x,y) gcd(t,n)=d Hence, for gcd(a, m) = 1,we have e m (aϑ n ) = d t d S(x,y) n S t/d (x/d,y) e m (aϑ dn ). T a (m,x,y) ( xd 1 y 1/2 (t/d) 11/32 m 5/16+o(1) + (t/d) 2 log 5 y ) d t xy 1/2 t 11/32 m 5/16+o(1) d t d 21/32 + t 2 log 5 y d t d 2. Estimating the first sum over the divisors d t as τ(t)= t o(1) and the second one as O(1), we complete the proof. In the case that m = q is a prime number, Lemma 5 can be used instead of Lemma 4 in the preceding proof to obtain the following bound: Theorem 12. Let q be a prime number, ϑ an integer not divisible by q, and t the multiplicative order of ϑ modulo q. Then the bound max { T a (q,x,y), T a (q,x,y) } xy 1/2 t 1/4 q 1/8+o(1) + t 2 log 5 y holds for all integers a such that q a, provided that t y/log 2 y. Accordingly, using Lemma 6, we derive the following result. Theorem 13. Let q be a prime number, ϑ an integer not divisible by q, and t the multiplicative order of ϑ modulo q. For every δ>0, there exist η>0 and κ>0 such that the bound 166 max { Ta (q,x,y), T a (q,x,y) } xq η + t 2 log 5 y

11 holds for all integers a such that q a, where the implied constant depends only on δ, provided that y q κ and t q δ. 4. REMARKS It would be natural to try to refine our results via one of the recursive identities commonly used in studying the distribution of smooth numbers. However it seems that such identities do not give any additional improvement in the case of exponential sums. One can easily improve the factor log x/log y in the bound of Theorem 10 by treating the corresponding divisor sum in a more intelligent manner. This however gives an improvement only when x is much larger than y, andy is much larger than q. But it is very plausible that in this range one can get a much better bound by simply using results about the uniformity of distribution of smooth numbers in arithmetic progressions; see [13,14]. We remark that Lemma 2 improves Theorem 3.2 of [1], which has the factor t 1/6 instead of the factor t 1/4 that appears here. Now, it is known that, for any fixed ϑ 2 and any function δ(q) 0, the inequality t q 1/2+δ(q) holds for almost all primes q; see [6]. In particular, we see from Lemma 3 that the Mersenne numbers M p = 2 p 1 with p x 2+δ are asymptotically uniformly distributed modulo q,for almost all primes q x. REFERENCES [1] Banks W., Conflitti A., Friedlander J.B., Shparlinski I.E. Exponential sums with Mersenne numbers, Compos. Math. 140 (2004) [2] Banks W., Conflitti A., Shparlinski I.E. Character sums over integers with restricted g-ary digits, Illinois J. Math. 46 (2002) [3] Bourgain J. Estimates on exponential sums related to Diffie Hellman distributions, C. R. Acad. Sci. Paris Ser (2004) [4] Bourgain J., Konyagin S.V. Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order, C. R. Acad. Sci. Paris Ser (2003) [5] Dewar M., Panario D., Shparlinski I.E. Distribution of exponential functions with k-full exponent modulo a prime, Indag. Math. 15 (2004) [6] Erdős P., Murty R. On the order of a(mod p), in: Proc. 5th Canadian Number Theory Association Conf., Amer. Math. Soc., Providence, RI, 1999, pp [7] Friedlander J.B., Hansen J., Shparlinski I.E. On character sums with exponential functions, Mathematika 47 (2000) [8] Friedlander J.B., Konyagin S.V., Shparlinski I.E. Some doubly exponential sums over Z m,acta Arith. 105 (2002) [9] Friedlander J.B., Shparlinski I.E. On the distribution of the power generator, Math. Comp. 70 (2001) [10] Friedlander J.B., Shparlinski I.E. Double exponential sums over thin sets, Proc. Amer. Math. Soc. 129 (2001) [11] Friedlander J.B., Shparlinski I.E. On the distribution of Diffie Hellman triples with sparse exponents, SIAM J. Discrete Math. 14 (2001) [12] Garaev M.Z. Double exponential sums related to Diffie Hellman distributions, Internat. Math. Res. Notices 17 (2005)

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