amsppt 140mm

On the exponential sum over r–free integers Antal Balog and Imre Z. Ruzsa A. Balog and I. Z. Ruzsa Exponential sum over r-free integers Research supported in part by HNFSR grant T 017433. Mathematical Institute, POBox 127, Budapest 1364, Hungary balog math-inst.hu ruzsa math-inst.hu

1. Introduction

Brüdern, Granville, Perelli, Vaughan and Wooley [3] studied bounds for the L^{1} mean of the exponential sum formed with the r–free integers, that is integers not divisible by any r-th power of a prime. They established that the L^{1} mean of this exponential sum is substantially smaller than the square root of the L^{2} mean, but still a positive power of the length of the exponential sum.

Let r ≥ 2 be fixed and a_{n} be the characteristic function of the r–free integers, that is

a_{n} = {1, if

n isr −−free, 0, otherwise, a_{0} = 0, a_{−n} = a_{n}.LetN beasufficientlylargeintegerandwedefineg_{1}(α) = ∑_{n=1}^{N} a_{n}e(nα), g_{2}(α) = ∑_{|n|≤N} a_{n}e(nα), g_{3}(α) = ∑_{|n|≤N} (1 − |n|N) a_{n}e(nα).T hemainresultof[2]statesthatN^{12r} ≪

∫_{0}^{1} |g_{1}(α)| dα ≪ N^{1r+1+ϵ}1.1forallpositive ϵ. The authors also expressed their opinion that the lower bound in (1.1) reflects the true order of magnitude.

In this paper we will show that the upper bound in (1.1) is almost best possible. We determine the precise order of magnitude.

THEOREM 1 We have

N^{1r+1} ≪ ∫_{0}^{1} |g_{1}(α)| dα ≪ N^{1r+1}. 1.2

The lower bound in (1.2) follows from the corresponding statement for g_{3}(α), namely

THEOREM 2 We have

N^{1r+1} ≪ ∫_{0}^{1} |g_{3}(α)| dα ≪ N^{1r+1}. 1.3

On the other hand, the upper bound in (1.3) follows from that in (1.2). As the definition of g_{3}(α) suggests, our basic tool is Fejér’s kernel

0 ≤ F (α) = ∑_{|n|≤N} (1 − |n|N) e(nα) = sin^{2} πNαN sin^{2} πα ≪ min(N, 1N∥α∥^{2}).1.4

More generally we will consider for any integers 1 ≤ K and 0 ≤ N

∑_{|n|≤N+K} min (1, N + K − |n|K) e(nα) == sin π(2N + K)α sin πKαK sin^{2} πα ≪ min(N+K, 1∥α∥, 1K∥α∥^{2}).1.5

The theorems are connected by the following relations:

|g_{2}(α)| ≤ 2|g_{1}(α)|, g_{3}(α) = ∫_{0}^{1} F (α − β)g_{2}(β) dβ.

Using ∫_{0}^{1} F (α) dα = 1 we arrive at

∫_{0}^{1} |g_{3}(α)| dα ≤ ∫_{0}^{1} ∫_{0}^{1} F (α − β)|g_{2}(β)| dβdα ≤ 2 ∫_{0}^{1} |g_{1}(β)| dβ.1.6

Therefore it is sufficient to prove the upper bound of Theorem 1, which is done in section 2, and the lower bound of Theorem 2, which is the content of section 3. In Section 4 we give an application to the exponential sum formed with the Möbius function.

2. Proof of the upper bound

We recall that a_{n} = ∑_{d^{r}|n} μ(d). Let 1 ≤ y < z be any real numbers. For n ≠ 0 we define c_{n} = c_{n}(y, z) as a middle part of the above sum, namely

c_{n} = ∑_{d^{r}|ny≤d<z} μ(d), 2.1

and we put c_{0} = 0.

LEMMA 1 For any 1 ≤ K < N and 1 ≤ y < z we have

∑_{N−K<n≤N} |c_{n}|^{2} ≪ Ky^{1−r} + N^{1/r} log^{3} z. 2.2

Proof We have

∑_{N−K<n≤N} |c_{n}|^{2} ≤ ∑_{N−K<n≤N}( ∑_{d^{r}|ny≤d<z} 1)^{2} = ∑_{y≤d_{1}<z} ∑_{y≤d_{2}<z} ∑_{N−K<n≤Nd_{1}^{r}|n , d_{2}^{r}|n} 1.2.3

The inner sum is bounded by K/[d_{1}, d_{2}]^{r} + 1 whenever [d_{1}, d_{2}] ≤ N^{1/r}, and empty otherwise. Thus, when it is not empty we have that 1 ≤ N^{1/r}/[d_{1}, d_{2}]. Therefore we can continue (2.3) by

≤ ∑_{y≤d_{1}<z} ∑_{y≤d_{2}<z} (K[d_{1}, d_{2}]^{r} + N^{1/r}[d_{1}, d_{2}]) ≤≤ ∑_{d_{1}≥y} ∑_{d_{2}≥y} Kd_{1}^{r}d_{2}^{r} ∑_{t|(d_{1},d_{2})} t^{r}+ ∑_{d_{1}<z} ∑_{d_{2}<z} N^{1/r}d_{1}d_{2} ∑_{t|(d_{1},d_{2})} t == K ∑_{t} t^{r} ( ∑_{d≥y , t|d} 1d^{r})^{2}+N^{1/r} ∑_{t<z} t ( ∑_{d<z , t|d} 1d)^{2} ≤≤ K ∑_{t} t^{−r} (min (1, (ty)^{r−1}))^{2}+N^{1/r} ∑_{t<z} 1t (∑_{d<z} 1d)^{2} ≪≪ K ∑_{t≤y} t^{r−2}y^{2r−2}+K ∑_{t>y} 1t^{r}+N^{1/r} log^{3} z ≪≪ Ky^{1−r}+N^{1/r} log^{3} z.2.4

This concludes the proof of the Lemma.

We continue with the upper estimates of Theorem 1.

Next note that for any integers 0 ≤ N, 1 ≤ d ≤ N + K and arbitrary M we have from (1.5) that

∑_{|n|≤N+Kn≡M(d)} min (1, N + K − |n|K) e(nα) == e(Mα) ∑_{|m|≤(N+K)/d} min (1, [N + Kd] − |m|[Kd]) e(mdα)+O(1) ≪≪ min(N + Kd, 1∥dα∥, dK∥dα∥^{2}).2.5

Let D = N^{1/(r+1)}, and let S be defined by 2^{S−1} < D ≤ 2^{S}. Note that S ≪ log N.

We decompose a_{n} as follows

a_{n} = ∑_{s=1}^{S} c_{n}(D2^{−s}, D2^{−s+1}) + c_{n}(D, N^{1/r}).2.6

We have

∫_{0}^{1} |∑_{n=1}^{N} a_{n}e(nα)| dα ≤≤ ∑_{s=1}^{S} ∫_{0}^{1} |∑_{n=1}^{N} c_{n}(D2^{−s}, D2^{−s+1})e(nα)| dα+∫_{0}^{1} |∑_{n=1}^{N} c_{n}(D, N^{1/r})e(nα)| dα.2.7

The Cauchy–Schwarz inequality, Parseval identity and Lemma 1 imply that

∫_{0}^{1} |∑_{n=1}^{N} c_{n}(D, N^{1/r})e(nα)| dα ≤≤ \sqrt{∑_{n≤N} |c_{n}(D, N^{1/r})|^{2}} ≪ \sqrt{ND^{1−r} + N^{1/r} log^{3} N} ≪ N^{1/(r+1)}.2.8

For a fixed s we abbreviate c_{n}(D2^{−s}, D2^{−s+1}) by c_{n} and we also introduce M = [N/2] and K = N2^{−rs}. We have

∑_{n=1}^{N} c_{n}e(nα) = ∑_{|n|≤M+K} min (1, 2M + K − |n|K) c_{M+n}e((M+n)α)−− ∑_{2M<n≤2M+K} 2M + K − |n|Kc_{n}e(nα)− ∑_{n≤K} K − |n|Kc_{n}e(−nα)+O(1).2.9

The contribution of the second and third sums can be handled much the same way as (2.8) yielding that

∫_{0}^{1} | ∑_{2M<n≤2M+K} 2M + K − |n|Kc_{n}e(nα)| dα ≤ \sqrt{∑_{2M<n≤2M+K} |c_{n}|^{2}} ≪≤ \sqrt{K(D2^{−s})^{1−r} + N^{1/r} log^{3} N} ≪ N^{1/(r+1)}2^{−s/2}+N^{1/(2r)} log^{3/2} N, 2.10

and the same for the contribution of the last sum. As for the first sum we use (2.5) and we get

∫_{0}^{1} | ∑_{|n|≤M+K} min (1, 2M + K − |n|K) c_{M+n}e((M+n)α)| dα ≤≤ ∑_{D2^{−s}≤d<D2^{−s+1}} ∫_{0}^{1} | ∑_{|n|≤M+Kn≡−M(d)} min (1, 2M + K − |n|K) e((M+n)α)| dα ≪≪ ∑_{D2^{−s}≤d<D2^{−s+1}} ∫_{0}^{1} min(Nd, 1∥dα∥, dK∥dα∥^{2}) dα ≪≪ ∑_{D2^{−s}≤d<D2^{−s+1}} ∫_{0}^{1} min(Nd, 1∥β∥, dK∥β∥^{2}) dβ ≪≪ ∑_{D2^{−s}≤d<D2^{−s+1}}(1+log NK) ≪ s2^{−s}D.2.11

Adding all terms from (2.8), (2.10) and (2.11) we get

∫_{0}^{1} |g_{1}(α)| dα ≪≪ ∑_{s=1}^{S}(s2^{−s}D+N^{1/(r+1)}2^{−s/2}+N^{1/(2r)} log^{3/2} N)+N^{1/(r+1)} ≪ N^{1/(r+1)}.2.12

3. The lower bound

We retain the notation of c_{n}(y, z) and D = N^{1/(r+1)} and we put

F_{h}(α) = ∑_{|n|≤Nh|n} (1 − |n|N) e(nα) = 1h ∑_{a=1}^{h} F (α − ah) ≥ 0.3.1

We start with the decomposition

g_{3}(α) = ∑_{|n|≤N} (1 − |n|N) a_{n}e(nα) == ∑_{|n|≤N} (1 − |n|N) c_{n}(1, D)e(nα)+ ∑_{|n|≤N} (1 − |n|N) c_{n}(D, N^{1/r})e(nα) == ∑_{d≤D} μ(d)F_{d^{r}}(α)− ∑_{d≤D} μ(d)+ ∑_{|n|≤N} (1 − |n|N) c_{n}(D, N^{1/r})e(nα).3.2