arXiv subjects

Daniele Mastrostefano

Publications and source records attributed to Daniele Mastrostefano.

9 recordsLinked to original sources

arXiv2021

An almost sure upper bound for random multiplicative functions on integers with a large prime factor

Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let $\varepsilon>0$ small. We prove that, as $x\rightarrow +\infty$, we almost surely have $$\bigg|\sum_{\substack{n\leq x\\ P(n)>\sqrt{x}}}f(n)\bigg|\leq\sqrt{x}(\log\log x)^{1/4+\varepsilon},$$ where $P(n)$ stands for the largest prime factor of $n$. This gives an indication of the almost sure size of the largest fluctuations of $f$.

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arXiv2021

A lower bound for the variance in arithmetic progressions of some multiplicative functions close to $1$

We investigate lower bounds for the variance in arithmetic progressions of certain multiplicative functions "close" to $1$. Specifically, we consider $α_N$-fold divisor functions, when $α_N$ is a sequence of positive real numbers approaching $1$ in a suitable way or $α_N=1$, and the indicator of $y$-smooth numbers, for suitably large parameters $y$. As a corollary, we will strengthen a previous author's result on the first subject and obtain matching lower bounds to some Barban-Davenport-Halberstam type theorems for $y$-smooth numbers. Incidentally, we will also find a lower bound for the variance in arithmetic progressions of the prime factors counting functions $ω(n)$ and $Ω(n)$.

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arXiv2020

On maximal product sets of random sets

For every positive integer N and every $α\in [0,1)$, let $B(N, α)$ denote the probabilistic model in which a random set $A\subset \{1,\dots,N\}$ is constructed by choosing independently every element of $\{1,\dots,N\}$ with probability $α$. We prove that, as $N\longrightarrow +\infty$, for every $A$ in $B(N, α)$ we have $|AA|\ \sim |A|^2/2$ with probability $1-o(1)$, if and only if $$\frac{\log(α^2(\log N)^{\log 4-1})}{\sqrt{\log\log N}}\longrightarrow-\infty.$$ This improves a theorem of Cilleruelo, Ramana and Ramaré, who proved the above asymptotic between $|AA|$ and $|A|^2/2$ when $α=o(1/\sqrt{\log N})$, and supplies a complete characterization of maximal product sets of random sets.

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arXiv2020

A lower bound for the variance of generalized divisor functions in arithmetic progressions

We prove that for a large class of multiplicative functions, referred to as generalized divisor functions, it is possible to find a lower bound for the corresponding variance in arithmetic progressions. As a main corollary, we deduce such a result for any $α$-fold divisor function, for any complex number $α\not\in \{1\}\cup-\mathbb{N}$, even when considering a sequence of parameters $α$ close in a proper way to $1$. Our work builds on that of Harper and Soundararajan, who handled the particular case of $k$-fold divisor functions $d_k(n)$, with $k\in\mathbb{N}_{\geq 2}$.

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arXiv2019

An Upper Bound for the Moments of a G.C.D. related to Lucas Sequences

Let $(u_n)_{n \geq 0}$ be a non-degenerate Lucas sequence, given by the relation $u_n=a_1 u_{n-1}+a_2 u_{n-2}$. Let $\ell_u(m)=lcm(m, z_u(m))$, for $(m,a_2)=1$, where $z_u(m)$ is the rank of appearance of $m$ in $u_n$. We prove that $$\sum_{\substack{m>x\\ (m,a_2)=1}}\frac{1}{\ell_u(m)}\leq \exp(-(1/\sqrt{6}-\varepsilon+o(1))\sqrt{(\log x)(\log \log x)}),$$ when $x$ is sufficiently large in terms of $\varepsilon$, and where the $o(1)$ depends on $u$. Moreover, if $g_u(n)=\gcd(n,u_n)$, we will show that for every $k\geq 1$, $$\sum_{n\leq x}g_u(n)^{k}\leq x^{k+1}\exp(-(1+o(1))\sqrt{(\log x)(\log \log x)}),$$ when $x$ is sufficiently large and where the $o(1)$ depends on $u$ and $k$. This gives a partial answer to a question posed by C. Sanna. As a by-product, we derive bounds on $#\{n\leq x: (n, u_n)>y\}$, at least in certain ranges of $y$, which strengthens what already obtained by Sanna. Finally, we start the study of the multiplicative analogous of $\ell_u(m)$, finding interesting results.

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arXiv2018

Positive proportion of short intervals containing a prescribed number of primes

We will prove that for every $m\geq 0$ there exists an $\varepsilon=\varepsilon(m)>0$ such that if $0<λ<\varepsilon$ and $x$ is sufficiently large in terms of $m$ and $λ$, then $$|\lbrace n\leq x: |[n,n+λ\log n]\cap \mathbb{P}|=m\rbrace|\gg_{m,λ} x.$$ The value of $\varepsilon(m)$ and the implicit constant on $λ$ and $m$ may be made explicit. This is an improvement of an author's previous result. Moreover, we will show that a careful investigation of the proof, apart from some slight changes, can lead to analogous estimates when considering the parameters $m$ and $λ$ to vary as functions of $x$ or restricting the primes to belong to specific subsets.

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arXiv2018

On numbers $n$ with polynomial image coprime with the $n$th term of a linear recurrence

Let $F$ be an integral linear recurrence, $G$ be an integer-valued polynomial splitting over the rationals, and $h$ be a positive integer. Also, let $\mathcal{A}_{F,G,h}$ be the set of all natural numbers $n$ such that $\gcd(F(n), G(n)) = h$. We prove that $\mathcal{A}_{F,G,h}$ has a natural density. Moreover, assuming $F$ is non-degenerate and $G$ has no fixed divisors, we show that $\mathbf{d}(\mathcal{A}_{F,G,1}) = 0$ if and only if $\mathcal{A}_{F,G,1}$ is finite.

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arXiv2018

Weighted Average Number of Prime $m$-tuples lying on an Admissible $k$-tuple of Linear Forms

We find an upper bound for the sum $\sum_{x<n\leq 2x}\textbf{1}_{\mathbb{P}}(n+h_{i_{1}})\cdots\textbf{1}_{\mathbb{P}}(n+h_{i_{m+1}})w_{n}$, where $(h_{i_{1}},...,h_{i_{m+1}})$ is any $(m+1)$-tuple of elements in the admissible set $\mathcal{H}=\{h_{1},...,h_{k}\}$, $m\geq 1$ and $x$ is sufficiently large, with the same weights $w_{n}$ used in the Maynard's paper "Dense clusters of primes in subsets". The estimate will be uniform over positive integer $k$ with $m+1\leq k\leq (\log x)^{1/5}$ and on admissible set $\mathcal{H}$ with $0\leq h_{1}<...<h_{k}\leq x$. The upper bound will depend on an integral of a smooth function and on the singular series of $\mathcal{H}$, which naturally arises in this context.

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arXiv2018

Short intervals containing a prescribed number of primes

We prove that for every nonnegative integer $m$ there exists an $\varepsilon>0$ such that if $λ\in (0,\varepsilon]$ and $x$ is sufficiently large in terms of $m$, then the number of positive integers $n\leq x$ for which the interval $[n,n+λ\log n]$ contains exactly $m$ primes is at least a constant times $x/\log x.$ This improves a result of T. Freiberg, when $λ$ is small.

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