Partial sums of biased random multiplicative functions

Abstract

Let P be the set of the primes. We consider a class of random multiplicative functions f supported on the squarefree integers, such that {f (p)} p∈P form a sequence of ±1 valued independent random variables with Ef (p) < 0, ∀p ∈ P. The function f is called strongly biased (towards classical Möbius function), if p∈P f (p) p = −∞ a.s., and it is weakly biased if p∈P f (p) p converges a.s. Let M f (x) := n≤x f (n). We establish a number of necessary and sufficient conditions for M f (x) = o(x 1−α ) for some α > 0, a.s., when f is strongly or weakly biased, and prove that the Riemann Hypothesis holds if and only if M fα (x) = o(x 1/2+ ) for all > 0 a.s., for each α > 0, where {fα}α is a certain family of weakly biased random multiplicative functions. 2010 Mathematics Subject Classification. Primary: 11N37. Secondary: 60F15.

On multiplicative functions that are small on average and zero-free regions for the Riemann zeta function

“…This result also improves the results in [1], in which it has been proved a result of same quality of Theorem 1.1 under randomness and bias assumptions.…”

“…) for any > 0, where λ is the Liouville function, i.e., λ is completely multiplicative and at primes λ(p) = −1. 1] is completely multiplicative and if its values at primes (f (p)) p are equal to β ∈ (−1, 1) \ {0} on average, then in the book of Tenenbaum [6] it has been developed a method -The Selberg-Delange method -which allow us evaluate the partial sums n≤x f (n). Indeed, if the Dirichlet series F (s) := ∞ n=1 f (n) n s satisfies a certain set of axioms, then the partial sums n≤x f (n) ∼ c f x (log x) 1−β , where c f = 0 is a constant which depends in f .…”

Random multiplicative functions: the Selberg-Delange class

“…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 G(s) = (s − s 0 ) m H(s).…”

“…Indeed, in the case that 1/2 < β < 1, one can check that the Dirichlet series of f β , say F β , satisfies the required set of axioms for the Selberg-Delange method in [11] to apply. The most difficult to check is an upper bound in vertical strips for a random Dirichlet series with independent and mean zero summands ∞ n=1 Xn n s , which has been done in [1]. Thus, the following holds almost surely…”

A note on prime number races and zero free regions for $L$ functions

“…Let P be the set of primes. In [1] it has been proved that under biased assumptions, i.e., if at primes (f (p)) p∈P is a sequence of independent random variables such that Ef (p) < 0 for all primes p, then the assumptions that f is small on average almost surely (a.s.) and F (1) = 0 a.s. imply that p∈P f (p)p −(1− ) converges for some > 0 a.s., and hence that F (s) = 0 for all Re(s) > 1 − , a.s.…”