Exponential and character sums with Mersenne numbers William D. Banks Dept. of Mathematics, University of Missouri Columbia, MO 65211, USA [email protected] John B. Friedlander Dept. of Mathematics, University of Toronto Toronto, Ontario M5S 3G3, Canada [email protected] Moubariz Z. Garaev Instituto de Matem´aticas Universidad Nacional Aut´onoma de M´exico C.P. 58089, Morelia, Michoac´an, M´exico [email protected] Igor E. Shparlinski Dept. of Computing, Macquarie University Sydney, NSW 2109, Australia [email protected] Dedicated to the memory of Alf van der Poorten 1 Abstract We give new bounds on sums of the form Λ(n) exp(2πiagn/m) and Λ(n)χ(gn + a), 6 6 nXN nXN where Λ is the von Mangoldt function, m is a natural number, a and g are integers coprime to m, and χ is a multiplicative character modulo m. In particular, our results yield bounds on the sums exp(2πiaMp/m) and χ(Mp) 6 6 pXN pXN p with Mersenne numbers Mp = 2 − 1, where p is prime. AMS Subject Classification Numbers: 11L07, 11L20. 1 Introduction Let m be an arbitrary natural number, and let a and g be integers that are coprime to m. Our aim in the present note is to give bounds on exponential sums and multiplicative character sums of the form n n Sm(a; N)= Λ(n)em(ag ) and Tm(χ, a; N)= Λ(n)χ(g + a), 6 6 nXN nXN where em is the additive character modulo m defined by em(x) = exp(2πix/m) (x ∈ R), and χ is a nontrivial multiplicative character modulo m. As usual, Λ denotes the von Mangoldt function given by log p if n is a power of the prime p, Λ(n)= (0 if n is not a prime power. 2 Let t denote the multiplicative order of g modulo m, i.e., the smallest natural number such that gt ≡ 1 (mod m). For the exponential sums defined above, Banks et al [1] have established the following bound as N →∞: max Sm(a; N) gcd(a,m)=1 11 5 5 5 7 Nt− 32 m 16 + N 6 t 48 m 24 N o(1) for all m > 1, (1) 6 − 1 1 5 2 1 o(1) (Nt 6 q 8 + N 6 t 9 q 6 N if m = q is prime. Furthermore, as previously observed in [2, Lemma 2], using a bound of Garaev [7] one can improve (1) for a prime m = q as follows: − 1 1 5 2 1 o(1) max Sq(a; N) 6 Nt 4 q 8 + N 6 t 9 q 6 N . (2) gcd(a,q)=1 We remark that an even stronger bound which is valid for almost all primes q has been obtained by Garaev and Shparlinski [10]. In this paper, using bounds for single and bilinear exponential sums from [1] and exploiting an idea of Garaev [8] to handle double sums over a certain “hyperbolic” region, we give improvements of both (1) and (2). As for the multiplicative character sums defined above, in an earlier work [3] we have presented a bound on Tm(χ, a; N) in the case that m = q is prime, but our proof contains a gap (see [3, Theorem 12]; the condition that the intervals of summation in our double sums contain distinct elements modulo t does not necessary hold). In the present note, although we do not completely recover [3, Theorem 12], we derive a bound that is nontrivial over a range that is only slightly shorter (also in the case of prime m = q). We note that, using our results in the case g = 2 together with partial summation, one obtains nontrivial bounds on the sums em(aMp) and χ(Mp) (3) p6N p6N p Xprime p Xprime p with Mersenne numbers Mp =2 − 1, where p is prime. In particular, we see from (1) and (2) that the sums Sm(a; N) admit a nontrivial estimate if, for some fixed ε> 0, we have t > m10/11+ε and N > t5/8m7/4+ε (4) 3 for general m, and we improve this to t > q1/2+ε and N > t4/3q1+ε (5) for the case of prime m = q. By comparison, our new estimates for Sm(a; N) given herein are, for gen- eral m, stronger than those given previously, although they are nontrivial only under the same condition (4). For the case of prime m = q they are not only stronger but also extend the region (5) to t > q1/2+ε and N > t1/2q5/4+ε. (6) Our bound on Tq(χ, a; N) is also nontrivial under the same condition (6). Throughout the paper, the implied constants in the symbol ‘≪’ may depend on the parameter ε (when present) but are absolute otherwise (we recall that the notation A ≪ B is equivalent to the assertion that |A| 6 cB for some constant c > 0). As a consequence, all of our results below are uniform in all parameters other than ε. In particular, our bounds are uniform over all integers a coprime to the modulus m and over all integers g with the same multiplicative order t modulo m. 2 Preparation 2.1 Vaughan’s bound We need the following result of Vaughan [15], which is stated here in the form given in [4, Chapter 24]. Lemma 1. For any complex-valued function f(n) and any real numbers U,V > 1 with UV 6 N, we have Λ(n)f(n) ≪ Σ1 + Σ2 + Σ3 + Σ4, 6 nXN 4 where Σ1 = Λ(n)f(n) , n6U X Σ2 = (log UV ) f(kℓ) , k6UV ℓ6N/k X X Σ3 = (log N) max f (kℓ) , w>0 k6V w<ℓ6N/k X X Σ4 = Λ(ℓ) µ(d) f(kℓ) . ! kℓ6N d | k k>V,X ℓ>U dX6V 2.2 Bounds on exponential sums As in [1] we need bounds on exponential sums with exponential functions over consecutive integers. The following statement is [1, Lemma 2.2]; it follows immediately from a result of Korobov [14, Theorem 10, Chapter 1] (see also the proof of [13, Lemma 2]). Lemma 2. Suppose that ϑ is coprime to m, and let T be the multiplicative order of ϑ modulo m. Then, for any H1 <H2 and any integer a coprime to m, we have n −1 1 +ε em(aϑ ) ≪ (H2 − H1)T +1 m 2 . H1<n6H2 X We also need the following bound on bilinear exponential sums with exponential functions, which is a special case of [1, Lemma 2.5]. Lemma 3. Suppose that g is coprime to m, and let t be the multiplicative order of g modulo m. Let K, L be natural numbers. Then, for any two K L sequences (αk)k=1 and (βℓ)ℓ=1 of complex numbers and any integer a coprime to m, we have kℓ − 1 1 − 1 1 21 5 +ε αk βℓ em(ag ) ≪ AB Kt 2 + K 2 Lt 2 + L 2 t 32 m 16 , k6K ℓ6L X X where A = max |αk| and B = max |βℓ|. (7) k6K ℓ6L 5 In the special case that m = q is a prime number, a stronger bound follows immediately from [1, Lemma 2.7], but one can do better using an improved version of that estimate due to Garaev and Karatsuba [9]. Lemma 4. Suppose that q ∤ g, and let t be the multiplicative order of g K modulo q. Let K, L be natural numbers. Then, for any two sequences (αk)k=1 L and (βℓ)ℓ=1 of complex numbers and any integer a not divisible by q, we have kℓ − 1 1 − 1 3 1 1 +ε αk βℓ eq(ag ) ≪ AB Kt 2 + K 2 Lt 4 + L 4 t 2 q 8 , k6K ℓ6L X X where A and B are defined by (7). Proof. In the case that K, L 6 t, an application of [9, Corollary 3] (with the choice k = 1) yields the bound kℓ 1 3 1 1 +ε αk βℓ eq(ag ) ≪ ABK 2 L 4 t 2 q 8 . 6 6 kXK Xℓ L For arbitrary values of K and L, we split the double sums into at most (Kt−1 +1)(Lt−1 + 1) double sums with at most min{K, t}· min{L, t} terms, deriving the bound kℓ −1 −1 1 3 1 1 +ε αk βℓ eq(ag ) ≪ (Kt + 1)(Lt + 1) min{K, t} 2 min{L, t} 4 t 2 q 8 . 6 6 kXK Xℓ L Since 1 1 1 (Kt−1 + 1) min{K, t} 2 6 Kt− 2 + K 2 and 3 1 3 (Lt−1 + 1) min{L, t} 4 6 Lt− 4 + L 4 , the result follows. ⊓⊔ Next, we use an idea of Garaev [8] to derive a variant of Lemma 3 in which the summation limits over ℓ depend on the parameter k. Lemma 5. Let the notation be as in Lemma 3. For any two sequences K K (Lk)k=1 and (Mk)k=1 of nonnegative integers such that Mk < Lk 6 L for each k, we have kℓ − 1 1 − 1 1 21 5 ε αk βℓ em(ag ) ≪ AB Kt 2 + K 2 Lt 2 + L 2 t 32 m 16 (mL) . k6K M <ℓ6L X kX k 6 Proof. For each inner sum we have kℓ kℓ 1 βℓ em(ag )= βℓ em(ag ) · eL(r(ℓ − s)) L 1 1 Mk<ℓ6Lk ℓ6L MK <s6Lk 6 X X X − 2 L<rX 2 L 1 kℓ = eL(−rs) βℓ eL(rℓ) em(ag ).