Arithmetic functions on Beatty sequences
Alex G. Abercrombie 3 Middle Row, Golden Hill Pembroke SA71 4TD, UK [email protected] 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 We study sums of the form
Sα(f,x)= f(n), 6 n x, n∈Bα where f is an arbitrary arithmetic function satisfying a mild growth condition, and Bα = (⌊αk⌋)k∈N is the homogeneous Beatty sequence corresponding to a real number α > 1. We show that for almost all α> 1 the asymptotic formula −1 Sα(f,x) ∼ α f(n) (x →∞) 6 n x holds, and we give a strong bound on the error term. Our results extend and improve the earlier results of several authors.
1 1 Introduction
1.1 Background For a real number α > 1, the homogeneous Beatty sequence corresponding to α is the sequence of natural numbers given by
Bα =(⌊αk⌋)k∈N, where ⌊t⌋ denotes the greatest integer 6 t. Beatty sequences appear in a variety of contexts and have been extensively explored in the literature. In particular, summatory functions of the form
Sα(f, x)= f(n) (1) n6x, n∈B α have been studied when the arithmetic function f is • a multiplicative or an additive function (see [1, 2, 8, 9, 10, 11]);
• a Dirichlet character (see [2, 3, 5]);
• the characteristic function of primes or smooth numbers (see [4, 6, 7]). For an arbitrary arithmetic function f we denote
S(f, x)= S1(f, x)= f(n). (2) n6x Abercrombie [1] has shown that for the divisor function τ the asymptotic formula −1 5/7+ε Sα(τ, x)= α S(τ, x)+ O(x ) (3) holds for any ε> 0 and almost all α> 1 (with respect to Lebesgue measure), where the implied constant depends only on α and ε. This result has been improved and extended by Zhai [14] as follows. For a fixed integer r > 1, let τr(n) be the number of ways to express n as a product of r natural numbers, expressions with the same factors in a different order being counted as different (in particular, τ2 = τ is the usual divisor function). In [14] it is shown that the asymptotic formula
−1 (r−1)/r+ε Sα(τr, x)= α S(τr, x)+ O(x ) (4)
2 holds for any ε> 0 and almost all α> 1 (in the special case r = 2 a similar result has also been obtained by Begunts [8]). The estimate (4) has been further improved by L¨uand Zhai [11] as follows:
O(x(r−1)/r+ε) if2 6 r 6 4; S (τ , x)= α−1S(τ , x)+ (5) α r r O(x4/5+ε) if r > 5. 1.2 Our result In this paper, we use the methods of [1] to derive an asymptotic formula for Sα(f, x) which holds for almost all α > 1 whenever f satisfies a rather mild growth condition. In particular, we do not stipulate any conditions on the multiplicative or additive properties of f (or on any other properties of f except for the rate of growth). Our general result, when applied to the divisor functions, yields a statement stronger than (3) and an improvement of (5) for all r > 4, and it can be applied to many other number theoretic functions (and to powers and products of such functions), including:
the M¨obius function µ(n),
the Euler function ϕ(n),
the number of prime divisors ω(n),
the sum σg(n) of the digits of n in a given base g > 2.
On the other hand, we note that although the results of [1, 11, 14] are formulated as bounds which hold for almost all α, the methods of those papers are somewhat more explicit than ours, and the results can be applied to any “individual” numbers α whose rational approximations satisfy certain hypotheses; thus, one can derive variants of (3), (4) and (5) for specific values of α (or over some interesting classes of α, such as the class of algebraic numbers).
1.3 Notation Throughout the paper, implied constants in the symbols O, ≪ and ≫ may depend (where obvious) on the parameters α,ε but are absolute otherwise. We recall that the notations U = O(V ), U ≪ V , and V ≫ U are all
3 equivalent to the assertion that the inequality |U| 6 cV holds with some constant c> 0. We also use t to denote the distance from t ∈ R to the nearest integer.
1.4 Acknowledgements During the preparation of this paper, I. S. was supported in part by ARC grant DP0556431.
2 Main Result
2.1 Formulation We denote −1 ∆α(f, x)= Sα(f, x) − α S(f, x) (6) and M(f, x)=1+max |f(n)| : n 6 x .
Theorem 1. For fixed ε> 0 and almost all real numbers α> 1, the following bound holds: 2/3+ε ∆α(f, x) ≪ x M(f, x).
2.2 Preparations
We follow the arguments of [1]. For any real number x > 1, let ψx be the trigonometric polynomial of Vaaler [13] given by
2πimt ψx(t)= ax(m) e (t ∈ R), 1/2 16|m |6x where for each integer m in the sum we put
πm (1 −|m |) cot(πm )+ |m | m a (m)= − x x x x with m = . (7) x 2πim x x1/2 +1 As in [1, Section 3] we note that the inequality
u(1 − u) cot(πu) 6 1 (0 6 u 6 1) 4 immediately implies the uniform bound 1 a (m) ≪ (1 6 |m| 6 x1/2). (8) x |m|
The function ψx is an exceptionally good approximation to the “sawtooth” function ψ(t) = {t} − 1/2, where {t} denotes the fractional part of t ∈ R. Indeed, by [1, Corollary 2.9] we have
csc2(πt) csc2(πt) ψ(t) − ψ (t) 6 ≪ . (9) x 2(x1/2 + 1)2 x To prove the theorem, we can clearly assume that α> 1 is irrational. In this case, one sees that a natural number n is a term in the Beatty sequence −1 Bα (that is, n = ⌊αk⌋ for some k ∈ N) if and only if α n lies in the set
t ∈ R : 1 − α−1 6 {t} < 1 . As the characteristic function ξα of that set satisfies the relation
−1 −1 ξα(t)= α + ψ(t) − ψ(t + α ) for every t ∈ R, it follows that
−1 f(n)= f(n) ξα(α n) 6 n6x n x, n∈Bα = f(n) α−1 + ψ(α−1n) − ψ(α−1(n + 1)) . n6x Taking into account the definitions (1), (2) and (6), we see that
∆α(f, x) 6 Qα(f, x) + |f(n)| Rα(n, x), (10) n6x where −1 −1 Qα(f, x)= f(n) ψx(α n) − ψx(α (n + 1)) , n6x and
−1 −1 −1 −1 Rα(n, x)= ψ(α n) − ψx(α n) + ψ(α (n + 1)) − ψx(α (n + 1)) . 5 2.3 Growth of the function Qα(f,x) We need the following estimate on the finite differences of the function Qα(f, x) which could be of independent interest. Lemma 1. For a fixed irrational α> 1 we have
Qα(f,y) − Qα(f, x) ≪ (y − x) M(f,y) (1 6 x 6 y 6 2x).
Proof. For any t ∈ R we have
ψy(t) − ψx(t)= S1 + S2, where
2πimt 2πimt S1 = (ay(m) − ax(m)) e and S2 = ay(m) e . 1/2 1/2 1/2 16|m |6x x <