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Marco Aymone

Publications and source records attributed to Marco Aymone.

At least 19 recordsLinked to original sources
arXiv2025

On the positivity of some weighted partial sums of a random multiplicative function

Inspired by the papers by Angelo and Xu, Q.J Math., 74, pp. 767-777, and improvements by Kerr and Klurman, arXiv:2211.05540, we study the probability that the weighted sums of a Rademacher random multiplicative function, $\sum_{n\leq x}f(n)n^{-σ}$, are positive for all $x\geq x_σ\geq 1$ in the regime $σ\to1/2^+$. In a previous paper by Heap, Zhao and the author, and by the author, when $0\leq σ\leq 1/2$ this probability is zero. Here we give a positive lower bound for this probability depending on $x_σ$ that becomes large as $σ\to1/2^+$. The main inputs in our proofs are a maximal inequality based in relatively high moments for these partial sums combined with a Bonami--Halász's moment inequality, and also explicit estimates for the partial sums of non-negative multiplicative functions.

math.NT↗
arXiv2025

On the expected number of roots of a random Dirichlet polynomial

Let $T>0$ and consider the random Dirichlet polynomial $S_T(t)=Re\, \sum_{n\leq T} X_n n^{-1/2-it}$, where $(X_n)_{n}$ are i.i.d. Gaussian random variables with mean $0$ and variance $1$. We prove that the expected number of roots of $S_T(t)$ in the dyadic interval $[T,2T]$, say $\mathbb{E} N(T)$, is approximately $2/\sqrt{3}$ times the number of zeros of the Riemann $ζ$ function in the critical strip up to height $T$. Moreover, we also compute the expected number of zeros in the same dyadic interval of the $k$-th derivative of $S_T(t)$. Our proof requires the best upper bounds for the Riemann $ζ$ function known up to date, and also estimates for the $L^2$ averages of certain Dirichlet polynomials.

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arXiv2025

Sign changes of the partial sums of a random multiplicative function III: Average

Let $V(x)$ be the number of sign changes of the partial sums up to $x$, say $M_f(x)$, of a Rademacher random multiplicative function $f$. We prove that the averaged value of $V(x)$ is at least $\gg (\log x)(\log\log x)^{-1/2-ε}$. Our new method applies for the counting of sign changes of the partial sums of a system of orthogonal random variables having variance $1$ under additional hypothesis on the moments of these partial sums. In particular, we extend to larger classes of dependencies an old result of Erdős and Hunt on sign changes of partial sums of i.i.d. random variables. In the arithmetic case, the main input in our method is the ``\textit{linearity}'' phase in $1\leq q\leq 1.9$ of the quantity $\log \mathbb{E} |M_f(x)|^q$, provided by the Harper's \textit{better than squareroot cancellation} phenomenon for small moments of $M_f(x)$.

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arXiv2025

$Ω$-bounds for the partial sums of some modified Dirichlet characters II

A modified Dirichlet character $f$ is a completely multiplicative function such that for some Dirichlet character $χ$, $f(p)=χ(p)$ for all but a finite number of primes $p\in S$, and for those exceptional primes $p\in S$, $|f(p)|\leq 1$. If $χ$ is primitive and for each $p\in S$ we have $|f(p)|=1$, we prove that $\sum_{n\leq x}f(n)=Ω((\log x)^{(|S|-3)/2})$. This makes progress on a Conjecture due to Klurman, Mangerel, Pohoata and Teräväinen, c.f. Trans. Amer. Math. Soc., 374 (2021), pp. 7967--7990. Our proof combines tools from Analytic Number Theory, Harmonic Analysis, Baker's Theory on linear forms in logarithms and Discrepancy bounds for sequences uniformly distributed modulo $1$.

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arXiv2024

Sign changes of the partial sums of a random multiplicative function II

We study two models of random multiplicative functions: Rademacher random multiplicative functions supported on the squarefree integers $f$, and Rademacher random completely multiplicative functions $f^*$. We prove that the partial sums $\sum_{n\leq x}f^*(n)$ and $\sum_{n\leq x}\frac{f(n)}{\sqrt{n}}$ change sign infinitely often as $x\to\infty$, almost surely. The case $\sum_{n\leq x}\frac{f^*(n)}{\sqrt{n}}$ is left as an open question and we stress the possibility of only a finite number of sign changes, with positive probability.

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arXiv2023

How many real zeros does a random Dirichlet series have?

Let $F(σ)=\sum_{n=1}^\infty \frac{X_n}{n^σ}$ be a random Dirichlet series where $(X_n)_{n\in\mathbb{N}}$ are independent standard Gaussian random variables. We compute in a quantitative form the expected number of zeros of $F(σ)$ in the interval $[T,\infty)$, say $\mathbb{E} N(T,\infty)$, as $T\to1/2^+$. We also estimate higher moments and with this we derive exponential tails for the probability that the number of zeros in the interval $[T,1]$, say $N(T,1)$, is large. We also consider almost sure lower and upper bounds for $N(T,\infty)$. And finally, we also prove results for another class of random Dirichlet series, e.g., when the summation is restricted to prime numbers.

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arXiv2023

Convolution of periodic multiplicative functions and the divisor problem

We study a certain class of arithmetic functions that appeared in Klurman's classification of $\pm 1$ multiplicative functions with bounded partial sums, c.f., Comp. Math. 153 (8), 2017, pp. 1622-1657. These functions are periodic and $1$-pretentious. We prove that if $f_1$ and $f_2$ belong to this class, then $\sum_{n\leq x}(f_1\ast f_2)(n)=\Omega(x^{1/4})$. This confirms a conjecture by the first author. As a byproduct of our proof, we studied the correlation between $\Delta(x)$ and $\Delta(\theta x)$, where $\theta$ is a fixed real number. We prove that there is a non-trivial correlation when $\theta$ is rational, and a decorrelation when $\theta$ is irrational. Moreover, if $\theta$ has a finite irrationality measure, then we can make it quantitative this decorrelation in terms of this measure.

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arXiv2023

$Ω$-bounds for the partial sums of some modified Dirichlet characters

We consider the problem of $Ω$ bounds for the partial sums of a modified character, \textit{i.e.}, a completely multiplicative function $f$ such that $f(p)=χ(p)$ for all but a finite number of primes $p$, where $χ$ is a primitive Dirichlet character. We prove that in some special circumstances, $\sum_{n\leq x}f(n)=Ω((\log x)^{|S|})$, where $S$ is the set of primes $p$ where $f(p)\neq χ(p)$. This gives credence to a corrected version of a conjecture of Klurman et al., Trans. Amer. Math. Soc., 374 (11), 2021, 7967-7990. We also compute the Riesz mean of order $k$ for large $k$ of a modified character, and show that the Diophantine properties of the irrational numbers of the form $\log p / \log q$, for primes $p$ and $q$, give information on these averages.

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arXiv2022

Complex valued multiplicative functions with bounded partial sums

We present a class of multiplicative functions $f:\mathbb{N}\to\mathbb{C}$ with bounded partial sums. The novelty here is that our functions do not need to have modulus bounded by $1$. The key feature is that they pretend to be the constant function $1$ and that for some prime $q$, $\sum_{k=0}^\infty \frac{f(q^k)}{q^k}=0$. These combined with other conditions guarantee that these functions are periodic and have sum equal to zero inside each period. Further, we study the class of multiplicative functions $f=f_1\ast f_2$, where each $f_j$ is multiplicative and periodic with bounded partial sums. We show an omega bound for the partial sums $\sum_{n\leq x}f(n)$ and an upper bound that is related with the error term in the classical Dirichlet divisor problem.

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arXiv2022

Multiplicative functions supported on the $k$-free integers with small partial sums

We provide examples of multiplicative functions $f$ supported on the $k$-free integers such that at primes $f(p)=\pm 1$ and such that the partial sums of $f$ up to $x$ are $o(x^{1/k})$. 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 $x$ are $o(x^{1/(k+\frac{1}{2})+ε})$, for all $ε>0$. This generalizes to the $k$-free integers the results of Aymone, `` A note on multiplicative functions resembling the {M}öbius function'', J. Number Theory, 212 (2020), pp. 113--121.

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arXiv2021

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

In this short note we prove the following result: If a completely multiplicative function $f:\mathbb{N}\to[-1,1]$ is small on average in the sense that $\sum_{n\leq x}f(n)\ll x^{1-δ}$, for some $δ>0$, and if the Dirichlet series of $f$, say $F(s)$, is such that $F(1)=0$, then we obtain that for any $ε>0$, $\sum_{p\leq x}(1+f(p))\log p\ll x^{1-δ+ε}$. Moreover, a necessary condition for the existence of such $f$ is that the Riemann zeta function $ζ(s)$ has no zeros in the half plane $Re(s)>1-δ$.

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arXiv2021

The Erdős discrepancy problem over the squarefree and cubefree integers

Let $g:\mathbb{N}\to\{-1,1\}$ be a completely multiplicative function, $μ$ be the Möbius function and $μ_2^2(n)$ be the indicator that $n$ is cubefree. We prove that $f=μ^2g$ and $f=μ_2^2g$ have unbounded partial sums. Our proofs are built upon Klurman and Mangerel's proof of Chudakov's conjecture, Klurman's work on correlations of multiplicative functions and Tao's resolution of the Erdős discrepancy problem.

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arXiv2021

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

Let $χ$ be a real and non-principal Dirichlet character, $L(s,χ)$ its Dirichlet $L$-function and let $p$ be a generic prime number. We prove the following result: If for some $0\leq σ<1$ the partial sums $\sum_{p\leq x}χ(p)p^{-σ}$ change sign only for a finite number of $x$, then there exists $ε>0$ such that $L(s,χ)$ has no zeros in the half plane $Re(s)>1-ε$.

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arXiv2020

Random multiplicative functions: The Selberg-Delange class

Let $1/2\leqβ<1$, $p$ be a generic prime number and $f_β$ be a random multiplicative function supported on the squarefree integers such that $(f_β(p))_{p}$ is an i.i.d. sequence of random variables with distribution $\mathbb{P}(f(p)=-1)=β=1-\mathbb{P}(f(p)=+1)$. Let $F_β$ be the Dirichlet series of $f_β$. We prove a formula involving measure-preserving transformations that relates the Riemann $ζ$ function with the Dirichlet series of $F_β$, 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 sums of $f_β$.

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arXiv2020

Law of the iterated logarithm for a random Dirichlet series

Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables with distribution $\mathbb P(X_1=1)=\mathbb P(X_1=-1)=1/2$. Let $F(σ)=\sum_{n=1}^\infty X_nn^{-σ}$. We prove that the following holds almost surely \begin{equation*} \limsup_{σ\to 1/2^+}\frac{F(σ)}{\sqrt{2\mathbb E F(σ)^2\log\log \mathbb E F(σ)^2}}=1. \end{equation*}

math.PR↗
arXiv2020

Bernoulli Hyperplane Percolation

We study a dependent site percolation model on the $n$-dimensional Euclidean lattice where, instead of single sites, entire hyperplanes are removed independently at random. We extend the results about Bernoulli line percolation showing that the model undergoes a non-trivial phase transition and proving the existence of a transition from exponential to power-law decay within some regions of the subcritical phase.

math.PR↗
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.

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