Multiplicative character sums with twice-differentiable functions William D. Banks Department of Mathematics University of Missouri Columbia, MO 65211 USA
[email protected] Igor E. Shparlinski Department of Computing Macquarie University Sydney, NSW 2109, Australia
[email protected] Abstract For a nontrivial multiplicative character χ modulo p, we bound character sums N X χ(⌊f (n)⌋) Sf (χ; N ) = n=1
taken on the integer parts of a real-valued, twice-differentiable function f whose second derivative decays at an appropriate rate. For the special case that f (x) = xη with some positive real number η, our bounds extend recent results of several authors.
Mathematics Subject Classification (2000): 11B50, 11L40.
1
1
Introduction
In this note, we study and bound character sums of the form Sf (χ; N) =
N X
χ(⌊f (n)⌋)
(N > 1),
n=1
where f is real-valued, twice-differentiable function, and χ is a nontrivial multiplicative character modulo a prime p. Here, we use ⌊t⌋ to denote the greatest integer not exceeding the real number t. To obtain a nontrivial bound on Sf (χ; N), we need to assume that the derivatives f ′ and f ′′ satisfy certain growth conditions. We apply a method that has been used in [1, 2, 4] to study Beatty sequences, which correspond to the case of a linear function f (x) = αx + β. Some aspects of our approach closely resemble the van der Corput method for estimating exponential sums; see [14, Section 8.3]. The results of the present note apply, in particular, to power functions f (x) = xη with η ∈ (1, 4/3); they imply that for every ε > 0 and any sufficiently large prime p, there is a positive integer n 6 p1/(4−2η)+o(1) such that ⌊nη ⌋ is a quadratic non-residue modulo p. For η in a shorter interval, stronger bounds (obtained by a different method) on the least n > 1 such that ⌊nη ⌋ is a quadratic non-residue are given in [6, 13, 19]. More precisely, if η ∈ (1, 12/11) then [13, Theorem 3] asserts that there is a positive integer −1/2 n 6 p3e /(24−16η)+o(1) such that ⌊nη ⌋ is a quadratic non-residue modulo p. In [19, Theorem 4] the range has been extended to η ∈ (1, 32/29) and the exponent 3e−1/2 /(24 − 16η) has been replaced with 9e−1/2 /(64 − 40η)) (which is better for values η not too close to 1; we refer to [19] for details). Finally, in [6] the exponent e−1/2 /(8 − 2η) has been obtained for η ∈ (1, 8/7). We note that all of the papers [6, 13, 19] outline possibilities of some further improvements, but these do not reach our range of η ∈ (1, 4/3). We do not see any reason to doubt that results of the same type are possible for all η ∈ (1, 2), but their proofs may require some new arguments. Our results not only hold over a wider range of values of the parameter η, but they also hold for a much larger class of functions f . Furthermore, our main result (see Theorem 6.1) below, gives a nontrivial bound on character sums Sf (χ; N). For example, in the case of f (x) = xη it is nontrivial in the range p1/(4−2η)+ε 6 N 6 p1/η (1) 2
for any ε > 0. In particular, this implis the uniformity of distribution of quadratic non-residues and primitive roots in the same intervals, while the results and methods of [6, 13, 19] apply only to the existence of quadratic non-residues and do not apply to primitive roots at all. An interesting feature of our method is that we use estimates for character sums over short intervals [K + 1, K + L] (where L is small relative to K) to derive estimates for sums over (longer) initial intervals [1, N], which are of primary interest. We remark that, by combining the techniques of this work with some ideas from [3], one can study the distribution of values of a variety of arithmetic functions on the sequence of values ⌊f (n)⌋, n = 1, 2, . . . (see also [5]). Acknowledgements. During the preparation of this paper, the second author was supported in part by ARC grant DP0556431.
2
Notation
Throughout the paper, any implied constants in the symbols O, ≪ and ≫ are absolute except where these symbols are subscripted with the list of parameters on which the implied constants depend. We recall that all of the notations U = O(V ), U ≪ V , and V ≫ U are equivalent to the assertion that the inequality |U| 6 c V holds for some constant c > 0. For any finite set S, we use #S to denote its cardinality, and we write e(t) = e2πit for every real number t.
3
Discrepancy of fractional parts of twicedifferentiable functions
Recall that the discrepancy D(N) of a sequence of (not necessarily distinct) real numbers x1 , . . . , xN ∈ [0, 1) is the quantity defined by V (I, N) D(N) = sup − |I| , N I⊆[0,1)
where the supremum is taken over all intervals I = [a, b) contained in [0, 1), V (I, N) is the number of positive integers n 6 N such that xn ∈ I, and |I| = b − a is the length of I. 3
The Erd˝os–Tur´an inequality (see [9, 17]) provides the essential link between the discrepancy of a sequence and its associated exponential sums: Lemma 3.1. For every integer M > 1, the discrepancy D(N) of a sequence of real numbers x1 , . . . , xN ∈ [0, 1) satisfies the uniform bound N M X X 1 1 1 D(N) ≪ e(jxn ) . + M N j=1 j n=1
The following statement is a consequence of the van der Corput bound for exponential sums with twice-differentiable functions:
Lemma 3.2. Suppose that f (x) is a real-valued, twice-differentiable function such that 0 < α 6 f ′′ (x) 6 αβ
(or
0 < α 6 −f ′′ (x) 6 αβ)
throughout the interval [X, Y ], where Y > X + 1. Then the discrepancy Df (X, Y ) of the sequence of fractional parts {f (n)} X 1). X 0 there exists δ > 0 such that the bound XX au bv χ(u + v) ≪ AB(#U)(#V) p−δ ε
u∈U v∈V
holds uniformly for all subsets U, V ⊆ Fp such that #U > p1/2+ε
#V > pε ,
and
(2)
all nonprincipal characters χ modulo p, and all complex numbers au and bv , where A = max |au | and B = max |bv |. u∈U
v∈V
Proof. By the H¨older inequality, we have for every integer k > 1: 2k 2k X X XX 6 A2k (#U)2k−1 b χ(u + v) a b χ(u + v) v u v u∈U v∈V
u∈U v∈V
2k
6 A (#U)
2k−1
k Y
X X
bvi χ(u + vi )
u∈Fp v1 ,...,vk i=1 w1 ,...,wk
X
= A2k (#U)2k−1
bwj χ(u + wj )
j=1
bv bw
(v,w)∈V k ×V k
k Y
X
χ(fv (u))χ(fw (u)),
u∈Fp
where the outer sum runs over pairs (v, w) of k-tuples of numbers from V, and for every v = (v1 , . . . , vk ) ∈ V k we put bv =
k Y
bvj
and
fv (u) =
j=1
k Y
(u − vj )
(u ∈ Fp ).
j=1
Therefore, 2k XX a b χ(u + v) u v u∈U v∈V
2k
2k
6 A B (#U)
2k−1
X
X . χ(f (u))χ(f (u)) v w
(3)
(v,w)∈V k ×V k u∈Fp
Let S1 be the set of pairs (v, w) ∈ V k × V k such that fv /fw = (g/h)n for some rational function g/h defined over an algebraic closure Fp of Fp , 5
where n > 2 is the order of χ. If (v, w) ∈ S1 , then considering the multisets hv1 , . . . , vk i and hw1, . . . , wk i associated to v and to w, respectively, it is easy to see that the number #{1 6 i 6 k : vi = v0 } − #{1 6 j 6 k : wj = v0 } is a multiple of n for every v0 ∈ V. Consequently, if v0 is a component of either v or w, then v0 occurs at least twice in the multiset hv1 , . . . , vk , w1 , . . . , wk i, and therefore, #{v0 ∈ V : v0 ∈ hv1 , . . . , vk , w1 , . . . , wk i} 6 k. From this it follows that k X #V (2k)j ≪ (#V)k , #S1 6 k j j=1 and we have X X ≪ p(#V)k . χ(f (u))χ(f (u)) v w k
(4)
(v,w)∈S1 u∈Fp
Now let S2 = (V k × V k ) \ S1 . For each pair (v, w) ∈ S2 , the rational function fv /fw is not of the form (g/h)n , hence we can use the Weil bound (see [18, Theorem 3 of Chapter 6]) to deduce that X X χ(fv (u))χ(fw (u)) ≪ p1/2 (#V)2k . (5) k
(v,w)∈S2 u∈Fp
Inserting (4) and (5) into (3), we derive the bound XX au bv χ(u + v) u∈U v∈V
≪ AB(#U)(#V) p(#U)−1 (#V)−k + p1/2 (#U)−1 k
1/(2k)
.
Choosing k = ⌈(2 ε)−1⌉ and taking into account (2), we obtain the stated result.
6
5
Character sums over short intervals
Our bound for Sf (χ; N) relies on bounds for more general sums of the form X Sf (χ; K, L) = χ(⌊f (n)⌋), K 0 and 2/3 < κ < 1. Let f (x) be a real-valued, twice-differentiable function such that log f ′′ (x) = −κ. x→∞ log x lim
Then, for all nonprincipal characters χ modulo p and all real numbers K, L that satisfy the inequalities (i) K κ−ε > L > K κ/2 pε , (ii) K 6 p1/(2−κ) , (iii) L > p1/2+ε , the uniform bound Sf (χ; K, L) ≪ Lp−δ ε,f
holds with some constant δ > 0 that depends only on ε and f . Proof. Put H = 2pε/2 . Our hypotheses imply that for all sufficiently large p the inequalities 0 < α 6 f ′′ (x) 6 αβ hold throughout the interval [K, K + L + H] with some real numbers α and β of size α = K −κ+o(1) and β = K o(1) (p → ∞). (6) Then, for any integers n ∈ (K, K + L] and h ∈ [0, H − 1] we have ′
f (n + h) − f (n) − hf (n) =
Z
n
7
n+h
f ′′ (u)(n + h − u) du,
and therefore, 0 6 f (n + h) − f (n) − hf ′ (n) 6 0.5h2 αβ 6 0.5H 2 αβ. Since ′
′
f (n) − f (K) =
Z
n
f ′′ (u) du,
K
we also have 0 6 hf ′ (n) − hf ′ (K) 6 h(n − K)αβ 6 HLαβ. Consequently, 0 6 f (n + h) − f (n) − hf ′ (K) 6 2HLαβ
(7)
if p is large enough. Now let ∆ = p−ε/4 with 0 < ∆ < 1/2, and put D = ⌈∆−1 ⌉. By the pigeonhole principle there is a number j0 ∈ {0, . . . , D − 1} such that the fractional part {hf ′ (K)} lies in the interval [j0 /D, (j0 + 1)/D) for at least H/D values of h ∈ [0, H − 1]. Let H be the set of all such integers h, and put ξ0 = j0 /D. Then, #H > H/D > 0.5H∆ > pε/4 ,
(8)
and {hf ′ (K) − ξ0 } 6 ∆
(h ∈ H).
(9)
Finally, let N be the set of integers n ∈ (K, K + L] such that {f (n) + ξ0 } 6 1 − ∆ − 2HLαβ.
(10)
Using (7), (9) and (10) one sees that the relation ⌊f (n + h)⌋ = ⌊f (n) + ξ0 ⌋ + ⌊hf ′ (K) − ξ0 ⌋ holds for every pair (n, h) ∈ N × H. Applying Lemma 3.2 together with (6) we have |#N − L| ≪ ∆L + HL2 αβ + Lα1/3 β 2/3 + α−1/2 ≪ Lp−ε/4 + L2 K −κ+o(1) pε/2 + LK −κ/3+o(1) + K κ/2+o(1) . 8
(11)
Using (i) we see that the bound |#N − L| ≪ Lp−δ1
(12)
ε,f
holds with some constant δ1 > 0 that depends only on ε and f . For every integer h > 0 we have X Sf (χ; K, L) = χ(⌊f (n + h)⌋) + O(h). K pε/4+o(1)
#V > (#N )po(1) > p1/2+ε+o(1) ,
and
and the weights au and bv are of size po(1) . Applying Lemma 4.1 and taking into account (12), we see that the bound W ≪ L(#H)p−δ2 +o(1) ε,f
holds with some constant δ2 > 0 that depends only on ε and f . Combining this bound with (13) we obtain the stated result.
6
Character sums over initial intervals
Theorem 6.1. Let κ and ε be fixed real numbers such that 2/3 < κ < 1
0 0, which depends only on κ and ε, the following inequalities hold for any N satisfying the lower bound of (14): 10
(i) N 1−κ/2 > p3c , (ii) N κ/2−c > p3c , (iii) N > p1/2+3c , (iv) N κ−c > p1/2+3c . Let c be fixed, and put R = N 1+c−κ log2 p
and
∆ = pc/R − 1.
Since
∆ = (1 + o(1))
c log p = N κ−c−1 (log p)−1+o(1) R
(p → ∞),
it follows that N κ−c−1 p−c 6 ∆ 6 N κ−c−1
(15)
if p is sufficiently large. In particular, ∆K 6 K κ−c
(K 6 N).
(16)
On the other hand, if K > Np−c then using (15) together with (i) and (ii) we see that ∆K 1−κ/2 p−c > K 1−κ/2 N κ−c−1 p−2c > N 1−κ/2 N κ−c−1 p−3c > 1; that is, (K > Np−c ).
∆K > K κ/2 pc
(17)
Similarly, using (15), (iii) and (iv) we have ∆K > KN κ−c−1 p−c > N κ−c p−2c > p1/2+c
(K > Np−c ).
(18)
Taking into account the inequalities (16), (17) and (18), we see that the hypotheses of Theorem 5.1 are satisfied (with ε replaced by c) with any K in the range Np−c 6 K 6 N and with L = ∆K; consequently, the uniform bound Sf (χ; K, ∆K) ≪ ∆Kp−δ1 (Np−c 6 K 6 N) (19) ε,f
holds with some constant δ1 > 0 that depends only on ε and f . 11
We now write X
Sf (χ; N) =
χ(⌊f (n)⌋) +
R X
X
χ(⌊f (n)⌋),
j=1 Kj
