arXiv subjects

Marco Aymone

Publications and source records attributed to Marco Aymone.

22 records · Page 2Linked to original sources

arXiv2020

Partial sums of random multiplicative functions and extreme values of a model for the Riemann zeta function

We consider partial sums of a weighted Steinhaus random multiplicative function and view this as a model for the Riemann zeta function. We give a description of the tails and high moments of this object. Using these we determine the likely maximum of $T \log T$ independently sampled copies of our sum and find that this is in agreement with a conjecture of Farmer--Gonek--Hughes on the maximum of the Riemann zeta function. We also consider the question of almost sure bounds. We determine upper bounds on the level of squareroot cancellation and lower bounds which suggest a degree of cancellation much greater than this which we speculate is in accordance with the influence of the Euler product.

math.NT↗

arXiv2019

A note on multiplicative functions resembling the Möbius function

We provide examples of multiplicative functions $f$ supported on the squarefree integers, such that on primes $f(p)=\pm1$ and such that $M_f(x):=\sum_{n\leq x} f(n)=o(\sqrt{x})$. Further, by assuming the Riemann hypothesis (RH) we can go beyond $\sqrt{x}$-cancellation.

math.NT↗

arXiv2019

Real zeros of random Dirichlet series

Let $F(σ)$ be the random Dirichlet series $F(σ)=\sum_{p\in\mathcal{P}} \frac{X_p}{p^σ}$, where $\mathcal{P}$ is an increasing sequence of positive real numbers and $(X_p)_{p\in\mathcal{P}}$ is a sequence of i.i.d. random variables with $\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=1/2$. We prove that, for certain conditions on $\mathcal{P}$, if $\sum_{p\in\mathcal{P}}\frac{1}{p}<\infty$ then with positive probability $F(σ)$ has no real zeros while if $\sum_{p\in\mathcal{P}}\frac{1}{p}=\infty$, almost surely $F(σ)$ has an infinite number of real zeros.

math.PR↗

arXiv2015

Partial sums of biased random multiplicative functions

Let $\mathcal{P}$ be the set of the primes. We consider a class of random multiplicative functions $f$ supported on the squarefree integers, such that $\{f(p)\}_{p\in\mathcal{P}}$ form a sequence of $\pm1$ valued independent random variables with $\mathbb{E} f(p)<0$, $\forall p\in \mathcal{P}$. The function $f$ is called strongly biased (towards classical Möbius function), if $\sum_{p\in\mathcal{P}}\frac{f(p)}{p}=-\infty$ a.s., and it is weakly biased if $\sum_{p\in\mathcal{P}}\frac{f(p)}{p} $ converges a.s. Let $M_f(x):=\sum_{n\leq x}f(n)$. 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 $M_{f_α}(x)=o(x^{1/2+ε})$ for all $ε>0$ a.s., for each $α>0$, where $\{f_α\}_α$ is a certain family of weakly biased random multiplicative functions.

math.NT↗