Abstract
Let 1∕2≤β<1, p be a generic prime number and fβ be a random multiplicative function supported on the squarefree integers such that (fβ(p))p is an i.i.d. sequence of random variables with distribution P(f(p)=−1)=β=1−P(f(p)=+1). Let Fβ be the Dirichlet series of fβ. We prove a formula involving measure-preserving transformations that relates the Riemann ζ function with the Dirichlet series of Fβ, for certain values of β, and give an application. Further, we prove that the Riemann hypothesis is connected with the mean behavior of a certain weighted partial sum of fβ.
Highlights
We say that f : N → C is a multiplicative function if f = f (n)f (m) for all nonnegative 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 = 0
Let 1/2 ≤ β < 1, p be a generic prime number and fβ be a random multiplicative function supported on the squarefree integers such that (fβ(p))p is an i.i.d. sequence of random variables with distribution P(f (p) = −1) = β = 1 − P(f (p) = +1)
We prove that the Riemann hypothesis is connected with the mean behavior of a certain weighted partial sum of fβ
Summary
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 (x1/2+ ), for all > 0. Any improvement of the type n≤x μ(n) = O(x1− ) 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 mean behavior of the partial sums of the Möbius function led Wintner [12] to investigate the behavior of a random model f for the Möbius function.
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