MULTIPLICATIVE FUNCTIONS SUPPORTED ON THE k-FREE INTEGERS WITH SMALL PARTIAL SUMS MARCO AYMONE, CAIO BUENO, KEVIN MEDEIROS Abstract. We provide examples of multiplicative functions f supported on the k- free integers such that at primes f(p) = ±1 and such that the partial sums of f up to x are o(x1=k). Further, if we assume the Generalized Riemann Hypothesis, then we can improve the exponent 1=k: There are examples such that the partial sums up to 1=(k+ 1 )+ x are o(x 2 ), for all > 0. 1. Introduction. We say that an integer n is k-free if n is not divisible by any prime power pk, where k ≥ 2 is an integer; We say that f : N ! C is multiplicative if f(nm) = f(n)f(m) for all integers n ≥ 1 and m ≥ 1 such that gcd(n; m) = 1, and we say that f is completely multiplicative if f(nm) = f(n)f(m) for all integers n ≥ 1 and m ≥ 1. Here the Vinogradov notation f(x) g(x) means that for some constant C > 0, jf(x)j ≤ Cjg(x)j, for all sufficiently large x > 0. P In this paper we are interested in the partial sums n≤x f(n) where f : N ! {−1; 0; 1g is a multiplicative function such that at primes f(p) = ±1 and supported on the k-free integers, i.e., functions f such that f(n) = 0 if n is not k-free. When we consider the case of the squarefree integers (the case k = 2), if f is multiplicative, supported on the squarefree integers and f(p) = −1 for all primes p, then f = µ, the M¨obiusfunction. A well known fact is that the Riemann hypothesis (RH) is equivalent to the square-root cancellation of the partial sums of the M¨obius P 1=2+ function µ, more precisely, n≤x µ(n) x , for all > 0. Up to this date, the true arXiv:2101.00279v1 [math.NT] 1 Jan 2021 magnitude of the partial sums of the M¨obiusfunction still unknown, even conditionally on RH; The unconditional upper bound obtained by the classical zero free region for P p the Riemann zeta function is n≤x µ(n) x exp(−c log x), and the best conditional upper bound is due to Soundararajan [14] which states that under RH, P µ(n) p p n≤x x exp( log x(log log x)14). These results for the M¨obiusfunction naturally raises the question of how small the partial sums of a multiplicative function supported on the squarefree integers can be. This led Wintner [18] to investigate a random model for the M¨obiusfunction which consists in randomizing the values f(p) = ±1 at primes p, and thus ending up with a 1 random multiplicative function. Wintner proved that a random multiplicative function P 1=2+ has partial sums n≤x f(n) x , for all > 0, almost surely, and also that these partial sums are not x1=2−, almost surely. Wintner's seminal paper led to many investigations on the partial sums of a random multiplicative function. Here we refer for the interested reader: [4], [5], [7], [8], [9] and [11]. The best upper bound up to date p P 2+ is due to Lau, Tenenbaum and Wu [11]: n≤x f(n) x(log log x) , for all > 0, almost surely; The best Ω bound is due to Harper [8]: The partial sums P f(n) are p n≤x not x=(log log x)5=2+, for all > 0, almost surely. By these results for a random multiplicative function we can naturally ask if there are examples of multiplicative functions f supported on the squarefree integers such that at primes f(p) = ±1, and such that its partial sums P f(n) have cancellation p n≤x beyond the x-cancellation. In [3], the first author exhibited examples of such functions p such that P f(n) x exp(−c(log x)1=4), and further, by assuming RH we can go n≤x p P 2=5+ much more beyond x-cancellation: There are examples such that n≤x f(n) x , for all > 0. Here we ask the same question over the k-free integers: Question. For which values of exponents α > 0 there exists a multiplicative func- tion f that is supported on the k-free integers, at primes f(p) = ±1 and such that P α n≤x f(n) x ? Here we give a reason to assume f(p) = ±1 over primes p. With this we exclude trivial solutions to our problem, because if f is supported on the squarefree integers (and hence supported on the k-free integers), and if we allow f(p) = 0 over primes, P by choosing f(p) such that p≤x jf(p)j x , for all > 0, then the partial sums P α n≤x f(n) x , for all α > 0. By assuming that f(p) = ±1 for all primes p, we guarantee that the set of the integers n ≥ 1 such that f(n) = ±1 has a positive lower density, since it contains the set of the squarefree integers which has density 6/π2. Moreover, with this condition, the event in which f has small partial sums means a non-trivial cancellation among the values f(n), as a consequence of the choice of the values f(pm), for primes p and powers m ≥ 1. Our first result states: Theorem 1.1. There exists a multiplicative function f : N ! {−1; 0; 1g supported on the k-free integers such that at primes p, f(p) = ±1, and such that X x1=k f(n) ; exp(c(log x)1=4) n≤x for some constant c > 0. 2 Thus, we can go beyond x1=k-cancellation when we consider k-free integers, and the exponent 1=k is critical in our results { If we assume the Generalized Riemann Hypothesis (GRH 1), then we can go beyond the exponent 1=k: Theorem 1.2. Assume the Generalized Riemann Hypothesis. Then there exists a mul- tiplicative function f : N ! {−1; 0; 1g supported on the k-free integers such that at primes p, f(p) = ±1, and such that X 1=(k+ 1 )+ f(n) x 2 ; n≤x for any > 0. Our examples in Theorems 1.1 and 1.2 are constructed in light of the extreme examples in the Erd¨osdiscrepancy theory for multiplicative functions. The original formulation of the Erd¨osdiscrepancy problem (EDP) is whether there exists an arithmetic function f : N ! {−1; 1g such that its partial sums over any homegenous arithmetic progression are uniformly bounded by some constant C > 0. As shown by Tao in [15], no such function exists, and moreover, we might say that the essence of the EDP is whether a completely multiplicative function f : N ! fz 2 C : jzj = 1g has bounded or unbounded partial sums, see [15]. A consequence of the solution of the EDP is that a completely multiplicative func- tion f : N ! {−1; 1g has unbounded partial sums. It is important to mention that if we allow completely multiplicative functions vanish over a finite subset of primes, then we might get bounded partial sums, for instance, any real and non-principal Dirichlet character χ. These Dirichlet characters are what we call the extreme examples in the Erd¨osdiscrepancy theory for multiplicative functions. Following Granville and Soundararajan [6], define the \distance" up to x between two multiplicative functions jfj; jgj ≤ 1 as !1=2 X 1 − Re(f(p)g(p)) (f; g; x) := ; D p p≤x where in the sum above p denotes a generic prime. Moreover, let D(f; g) := D(f; g; 1), and we say that f pretends to be g, or that f is g-pretentious if D(f; g) < 1. In the resolution of the EDP (see Remark 3.1 of [15] and Proposition 2.1 of [2]), it has been shown that a multiplicative function f : N ! [−1; 1] with bounded partial 1We recall that ζ stands for the classical Riemann zeta function and for a Dirichlet character χ, P1 χ(n) L(s; χ) stands for the classical L-function L(s; χ) = n=1 ns . The Riemann Hypothesis for ζ states that all zeros ζ(s0) = 0 with real part Re(s0) > 0 have real part equals 1=2. The Generalized Riemann Hypothesis is the same statement for L(s; χ). 3 sums must be χ-pretentious, for some real Dirichlet character χ : N ! {−1; 0; 1g, 2 1 Pp jf(n)j provided that the logarithmic average log x x≤n≤x n 1. Thus, if we seek for multiplicative functions supported on the k-free integers with small partial sums, then we must look firstly to those that are χ-pretentious for some real and non-principal Dirichlet character χ. In our particular case, to construct examples for theorems 1.1 and 1.2 we assume much more than χ-pretentiousness. Indeed, we deduce the Theorem 1.1 from the following result: 2 Theorem 1.3. Let µk(n) 2 f0; 1g be the indicator that n is k-free. Let g : N ! {−1; 1g be a completely multiplicative function that it is χ-pretentious for some real and non- principal Dirichlet character χ, and further assume that X x1=k (1) j1 − g(p)χ(p)j p : exp(c log x) p≤x 2 Let f = µkg be a multiplicative function supported on the k-free integers.