Random multiplicative functions: The Selberg-Delange class

Let $1/2\leq\beta<1$, $p$ be a generic prime number and $f_\beta$ be a random multiplicative function supported on the squarefree integers such that $(f_\beta(p))_{p}$ is an i.i.d. sequence of random variables with distribution $\mathbb{P}(f(p)=-1)=\beta=1-\mathbb{P}(f(p)=+1)$. Let $F_\beta$ be the Dirichlet series of $f_\beta$. We prove a formula involving measure-preserving transformations that relates the Riemann $\zeta$ function with the Dirichlet series of $F_\beta$, for certain values of $\beta$, and give an application. Further, we prove that the Riemann hypothesis is connected with the mean behavior of a certain weighted partial sums of $f_\beta$.


Introduction.
We say that f : N → C is a multiplicative function if f (nm) = f (n)f (m) for all non-negative integers n and m with gcd(n, m) = 1, and that f has support on the squarefree integers if for any prime p and any integer power k ≥ 2, f (p k ) = 0.
An important example of such function is the Möbius µ: The multiplicative function supported on the squarefree integers such that at each prime p, µ(p) = −1.
Many important problems in Analytic Number Theory can be rephrased in terms of the mean behavior of the partial sums of multiplicative functions. For instance, the Riemann hypothesis -The statement that all the non-trivial zeros of the Riemann ζ function have real part equal to 1/2 -is equivalent to the statement that the partial sums of the Möbius function have square root cancellation, that is, n≤x µ(n) is O(x 1/2+ ), for all > 0. In this direction, the best result up to date is of the type n≤x µ(n) = O(x exp(−c(log x) α )), for some positive constant c > 0, and some 0 < α < 1. Any improvement of the type n≤x µ(n) = O(x 1− ) for some > 0 would be a huge breakthrough in Analytic Number Theory, since it would imply that the Riemann ζ function has no zeros with real part greater than 1 − . This equivalence between the Riemann hypothesis with the behavior of the partial sums of the Möbius function led Wintner [10]  have that f (n) is a random multiplicative function supported on the squarefree integers such that at primes p ∈ P (here P stands for the set of primes), (f (p)) p∈P is an i.i.d.
sequence of random variables whith distribution P(f (p) = −1) = P(f (p) = +1) = 1/2. It is important to obeserve that the sequence (f (n)) n∈N is highly dependent, for instance, since 30 = 2 × 3 × 5, we have that f (30) depends on the values f (2), f (3) and f (5). Wintner proved the square root cancellation for the partial sums of f , that is, n≤x f (n) = O(x 1/2+ ) for all > 0, almost surely, and hence the assertion that the Riemann hypothesis is almost always true. This upper bound has been improved several times: [3], [5], [2] and [7]. The best upper bound up to date is due to Lau, Tenenbaum and Wu [7], which states that n≤x f (n) = O( √ x(log log x) 2+ ) for all > 0, almost surely, and the best Ω result is due to Harper [6] which states that for any A > 5/2, Here we consider a slight different model for the Möbius function. We start with a parameter 1/2 ≤ β ≤ 1 and consider a random multiplictive function f β supported on the squarefree integers where at primes, (f β (p)) p∈P is an i.i.d. sequence of random variables with P(f β (p) = −1) = β = 1 − P(f β (p) = +1). For β = 1/2 we recover the Wintner's model; For β = 1, f 1 is the Möbius µ; And for β < 1, we have that f β (n) equal to µ(n) with high probability as β is taken to be close to 1. In this paper we are interested in the following questions. Considering the first question, observe that Ef (p) = 1 − 2β, and thus, we might say that at primes, f β (p) is equal to 1 − 2β in average. In the case 1/2 < β < 1 the partial sums n≤x f β (n) are well understood by the Selberg-Delange method, see the book of Tenenbaum [9] chapter II.5 or the recent treatment of Granville and Koukoulopoulos [4]. Indeed, by the main result of [4], we have that for 1/2 < β < 1, the following holds almost surely where c f β is a random constant which is positive almost surely. In particular, this implies that n≤x f β (n) is not O(x 1−δ ), for any δ > 0, almost surely. This answers negatively our question 1.
Here we provide a different proof of a negative answer to our question 1 for certain values of β, that is, the statement that we do not have square root cancellation for n≤x f β (n) for certain values of β, almost surely. Further, by considering the question 2, we show that the Riemann hypothesis is equivalent to the square root cancellation of a certain weighted partial sums of f β .
Before we state our results, let us introduce some notation. Given a probability space (Ω, F, P), let ω be a generic element of Ω, and T : Ω → Ω be a measurepreserving transformation, i.e., P(T −1 (A)) = P(A), for all A ∈ F. We see the random multiplicative function f β defined over the probability space (Ω, F, P) as a function n s , is a random analytic function defined over the half plane H 1 := {s ∈ C : Re(s) > 1}, that is for all ω ∈ Ω.
n s . Then there exists a measure-preserving transformation T : Ω → Ω such that T 2 n = identity and such that the following formula holds for all Re(s) > 1 and all ω ∈ Ω: In particular, if β = 3/4, we have Corollary 1.1. For an integer n ≥ 1 and β = 1 − 1 2 n+1 , we have that for any δ > 0, Here we outline our proof of the Corollary 1.

Notation.
Here we let p denote a generic prime number and P to be the set of primes. We use f (x) g(x) and f (x) = O(g(x)) whenever there exists a constant c > 0 such that |f (x)| ≤ c|g(x)|, for all x in a certain set X -This set X could be all the interval x ∈ [1, ∞) or x ∈ (a − δ, a + δ), a ∈ R, δ > 0. We say that f (x) = o(g(x)) if lim f (x) g(x) = 0. The notation d|n means that d divides n. Here * stands for the Dirichlet convolution (f * g)(n) := d|n f (d)g(n/d). We denote d(n) = p|n 1, that is, the quantity of distinct primes that divides n. For a set A, 1 A (x) stands for the indicator function of the set A, that is,

3.1.
Construction of the probability space. We let P be the set of primes, Ω = [0, 1] P = {ω = (ω p ) p∈P : ω p ∈ [0, 1] for all p}, F the Borel sigma algebra of Ω and P be the product of Lebesgue measures in F. We set f β (p) as It follows that (f β (p)) p∈P are i.i.d. with distribution P(f β (p) = −1) = β = 1−P(f β (p) = +1). Also, we say that f β are uniformly coupled for different values of β.

3.2.
Construction of the measure-preserving transformation. Now if β = 1 − 1 2 n+1 with n ≥ 1 an integer, we partionate the interval [1/2, 1] into 2 n subintervals I k = (a k−1 , a k ] of lenght 1 2 n+1 and with endpoints a k = 1 2 + k 2 n+1 . It follows that a 0 = 1/2, a 2 n −1 = β and a 2 n = 1. Let T p : [0, 1] → [0, 1] be the following interval exchange transformation: For ω p ∈ [0, 1/2], T p (ω p ) = ω p ; In each interval I k as above the restriction T p | I k is a translation; T p (I 1 ) = I 2 n and for k ≥ 2, T p (I k ) = I k−1 . It follows that the kth iterate T k p (I k ) = I 2 n and T 2 n is the identity. Also, for each prime p, T p and its iterates preserve the Lebesgue measure and hence, T : Ω → Ω defined by T ω := (T p (ω p )) p∈P preserves P, and so its iterates.
Proof. We let F β be the Dirichlet series of f β and I k = (a k−1 , a k ] be as above. Notice that a 0 = 1/2 and a 2 n = 1, and hence F a 0 = F 1/2 and F a 2 n = F 1 = 1 ζ . Observe that Now, by the Euler product formula (2), we have that for all Re(s) > 1 Thus, as all I k have same lenght, we see that each is equal in probability distribution to the last Thus which concludes the proof.

Proof.
A standard result about Dirichlet series, is that the Dirichlet series of an arithmetic function f , say F (s), is the Mellin transform of the partial sums of f . Indeed, we have that for s in the half plane of convergence of F (s), Thus, we can conclude that the event in which the partial sums using Taylor expansion of each logarithm, we see that where A β (s) = O σ 0 (1) for all Re(s) ≥ σ 0 > 1/2. Since Ef β (p) = 1 − 2β < 0 for all primes p, we have by the Kolmogorv two series Theorem that lim s→1 + p∈P f β (p) p s = −∞ almost surely, and hence, lim s→1 + F β (s) = 0 almost surely.
If T is the meausre-preserving transformation as in Theorem 1.1, then the same is almost surely true for F β (s, T k ω). Further, in the Wintner's proof [10] of the square root cancellation of n≤x f 1/2 (n), it has been proved that F 1/2 (s) is almost surely a non-vanishing analytic function over the half plane {Re(s) > 1/2}. Indeed, this can be proved by the formula (3).
A well known fact is that the Riemann ζ function has a simple pole at s = 1, and hence, 1 ζ(s) has a simple zero at the same point. Moreover we recall that if an analytic function G has a zero at s = s 0 , then there exists a non-vanishing analytic function H at s = s 0 and a non-negative integer m, called the multiplicity of the zero s 0 , such that has a zero of multiplicity 2 n − 1 at s = 1 while the right side of the same equation has a zero of multiplicity at least 2 n at the same point, almost surely, which is a contradiction.
Thus we see that the probability of the event in which F β (s) has analytic continuation to Re(s) > 1 − δ is strictly less than one. Now we can check by the Euler product formula (2) that the event in which F β has analytic continuation to Re(s) > 1 − δ is a tail event for δ < 1, i.e., whether F β has analytic continuation to {Re(s) > 1 − δ} does not depend in any outcome of a finite number of random variables {f β (p) : p ≤ y}.

Indeed, we can write
is a non-vanishing analytic function in Re(s) > 0, we obtain that F β (s) has analytic continuation to Re(s) > 1 − δ (δ < 1) if and only if X y (s) := p>y 1 + f β (p) p s has analytic continuation to the same half plane. Since X y (s) is independent of {f β (p) : p ≤ y, p ∈ P} and the random variables (f β (p)) p∈P are independent, we conclude that the event in which F β has analytic continuation to {Re(s) > 1 − δ} is a tail event.
Thus by the Kolmogorov zero or one law, we have that the probability in which