Bull. Austral. Math. Soc. 78 (2008), 327–334 doi:10.1017/S0004972708000853 SUMS OF MULTIPLICATIVE FUNCTIONS OVER A BEATTY SEQUENCE AHMET M. GULO¨ GLU˘ ˛ and C. WESLEY NEVANS (Received 3 March 2008) Abstract We study sums involving multiplicative functions that take values over a nonhomogenous Beatty sequence. We then apply our result in a few special cases to obtain asymptotic formulas for quantities such as the number of integers in a Beatty sequence that are representable as a sum of two squares up to a given magnitude. 2000 Mathematics subject classification: 11E25, 11B83. Keywords and phrases: sums of multiplicative functions, Beatty sequences. 1. Introduction Let A ≥ 1 be an arbitrary constant, and let FA be the set of multiplicative functions f such that j f .p/j ≤ A for all primes p and X 2 2 j f .n/j ≤ A N .N 2 N/: (1) n≤N Exponential sums of the form X Sα, f .N/ D f .n/e.nα/ (α 2 R; f 2 FA/; (2) n≤N where e.z/ D e2πiz for z 2 R, occur frequently in analytic number theory. Montgomery and Vaughan have shown (see [8, Corollary 1]) that the upper bound N N.log R/3=2 S f .N/ A C (3) α, log N R1=2 −2 holds uniformly for all f 2 FA, provided that jα − a=qj ≤ q with some reduced fraction a=q for which 2 ≤ R ≤ q ≤ N=R. In this paper, we use the Montgomery– Vaughan result to estimate sums of the form c 2008 Australian Mathematical Society 0004-9727/08 $A2:00 C 0:00 327 Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.126, on 29 Sep 2021 at 13:36:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972708000853 328 A. M. Gulo¨ glu˘ and C. W. Nevans [2] X Gα,β; f .N/ D f .n/; (4) n≤N n2Bα,β where α, β 2 R with α > 1, f 2 FA, and Bα,β is the nonhomogenous Beatty sequence defined by Bα,β D fn 2 N V n D bαm C βc for some m 2 Zg: Our results are uniform over the family FA and nontrivial whenever log N X lim f .n/ D 1; N!1 N N log log n≤N a condition which guarantees that the error term in Theorem 1 is smaller than the main term. One can remove this condition, at the expense of losing uniformity with respect to f , and still obtain Theorem 1 for any bounded arithmetic function f (not necessarily multiplicative) for which the exponential sums in (2) satisfy X Sα, f .N/ D o f .n/ .N ! 1/: n≤N The general problem of characterizing functions for which this relation holds appears to be rather difficult; see [1] for Bachman’s conjecture and his related work on this problem. We shall also assume that α is irrational and of finite type τ. For an irrational number γ , the type of γ is defined by n t o τ D sup t 2 R V lim inf n γ n D 0 ; n!1 J K where · denotes the distance to the nearest integer. Dirichlet’s approximation theoremJ Kimplies that τ ≥ 1 for every irrational number γ . According to theorems of Khinchin [6] and Roth [10], τ D 1 for almost all real numbers (in the sense of the Lebesgue measure) and all irrational algebraic numbers γ , respectively; also see [2, 11]. Our main result is the following theorem. THEOREM 1. Let α, β 2 R with α > 1, and suppose that α is irrational and of finite type. Then, for all f 2 FA, X N log log N G .N/ D α−1 f .n/ C O ; α,β; f N n≤N log where the implied constant depends only on α and A. The following corollaries are immediate applications of Theorem 1. Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.126, on 29 Sep 2021 at 13:36:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972708000853 [3] Sums with multiplicative functions over a Beatty sequence 329 COROLLARY 2. The number of integers not exceeding N that lie in the Beatty sequence Bα,β and can be represented as a sum of two squares is CN N log log N #fn ≤ N V n 2 Bα,β ; n D C g D C O ; αplog N log N where Y C D 2−1=2 .1 − p−2/−1=2 D 0:76422365 ::: (5) p≡3 mod 4 is the Landau–Ramanujan constant. To state the next result, we recall that an integer n is said to be k-free if pk - n for every prime p. COROLLARY 3. For every k ≥ 2, the number of k-free integers not exceeding N that lie in the Beatty sequence Bα,β is N log log N #fn ≤ N V n 2 B ; n is k-freeg D α−1ζ −1.k/N C O ; α,β log N where ζ.s/ is the Riemann zeta function. Finally, we consider the average value of the number of representations of an integer from a Beatty sequence as a sum of four squares. COROLLARY 4. Let r4.n/ denote the number of representations of n as a sum of four squares. Then X π 2 N 2 N 2 log log N r .n/ D C O ; 4 N n≤N 2α log n2Bα,β where the implied constant depends only on α. Any implied constants in the symbols O and may depend on the parameters α and A but are absolute otherwise. We recall that the notation X Y is equivalent to X D O.Y /. 2. Preliminaries 2.1. Discrepancy of fractional parts We define the discrepancy D.M/ of a sequence of real numbers b1; b2;:::; bM 2 T0; 1/ by V.I; M/ D.M/ D sup − jIj ; (6) I⊆[0;1/ M where the supremum is taken over all possible subintervals I D .a; c/ of the interval T0; 1/, V.I; M/ is the number of positive integers m ≤ M such that bm 2 I, and jIj D c − a is the length of I. Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.126, on 29 Sep 2021 at 13:36:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972708000853 330 A. M. Gulo¨ glu˘ and C. W. Nevans [4] If an irrational number γ is of finite type, we let Dγ,δ.M/ denote the discrepancy of f C g M the sequence of fractional parts . γ m δ /mD1. By [7, Theorem 3.2, Ch. 2], we have the following result. LEMMA 5. For a fixed irrational number γ of finite type τ and for all δ 2 R, −1/τCo.1/ Dγ,δ.M/ ≤ M .M ! 1/; where the function defined by o.·/ depends only on γ . 2.2. Numbers in a Beatty sequence The following result is standard in characterizing the elements of the Beatty sequence Bα,β . − − LEMMA 6. Let α, β 2 R with α > 1, and set γ D α 1 and δ D α 1.1 − β/. Then n D bαm C βc for some m 2 Z if and only if 0 < fγ n C δg ≤ γ . From Lemma 6, an integer n lies in Bα,β if and only if n ≥ 1 and (γ n C δ/ D 1, where is the periodic function with period one whose values on the interval .0; 1U are given by 1 if 0 < x ≤ γ ; .x/ D 0 if γ < x ≤ 1: We wish to approximate by a function whose Fourier series representation is well behaved. This will give rise to the aforementioned exponential sum Sα, f .N/. To this end, we use the result of Vinogradov (see [15, Ch. I, Lemma 12]) which states that for any 1 such that 1 ≤ 1 f − g 0 < 1 < 8 and 1 2 min γ; 1 γ ; there exists a real-valued function 9 with the following properties: (i) 9 is periodic with period one; (ii) 0 ≤ 9.x/ ≤ 1 for all x 2 R; (iii) 9.x/ D .x/ if 1 ≤ fxg ≤ γ − 1 or if γ C 1 ≤ fxg ≤ 1 − 1; (iv) 9 can be represented by a Fourier series X 9.x/ D g.k/e.kx/; k2Z where g.0/ D γ and the Fourier coefficients satisfy the uniform bound g.k/ minfjkj−1; jkj−21−1g .k 6D 0/: (7) 3. Proofs 3.1. Proof of Theorem 1 Using Lemma 6, we rewrite the sum (4) in the form X Gα,β; f .N/ D f .n/ (γ n C δ/: n≤N Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.126, on 29 Sep 2021 at 13:36:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972708000853 [5] Sums with multiplicative functions over a Beatty sequence 331 Replacing by 9, we obtain X X Gα,β; f .N/ D f .n/9(γ n C δ/ C O f .n/ ; (8) n≤N n2V .1;N/ where V .1; N/ is the set of positive integers n ≤ N for which fγ n C δg 2 T0; 1/ [ (γ − 1; γ C 1/ [ .1 − 1; 1/: Since the length of each interval above is at most 21, it follows from definition (6) and Lemma 5 that jV .1; N/j 1N C N 1−1=.2τ/; where we have used the fact that α and γ have the same type τ.