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Alessandro Gambini

Publications and source records attributed to Alessandro Gambini.

15 recordsLinked to original sources
arXiv2026

On the discrete convolution of the Liouville and M\"obius functions

In this article we study some properties of the discrete convolution of Liouville function $S(n):=\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)$, which is a Goldbach-type counting function of representations. In particular, using the general approach introduced in a recent paper \cite{CGZ}, we will give an explicit formula for weighted averages of $S(n)$ with a general weights $f(w)$ that verify suitable conditions. This formula allows us to obtain interesting information about the Dirichlet and power series of $S(n)$ and the discrete convolution with an arbitrary numbers of factors $\lambda(n)$.

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arXiv2025

On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers

We provide numerical evidence towards three conjectures on harmonic numbers by Eswarathasan--Levine and Boyd. Let $J_p$ denote the set of integers $n\geq 1$ such that the harmonic number $H_n$ is divisible by a prime $p$. The conjectures state that: $(i)$ $J_p$ is always finite and of the order $O(p^2(\log\log p)^{2+\epsilon})$; $(ii)$ the set of primes for which $J_p$ is minimal (called harmonic primes) has density $e^{-1}$ among all primes; $(iii)$ no harmonic number is divisible by $p^4$. We prove $(i)$ and $(iii)$ for all $p\leq 16843$ with at most one exception, and enumerate harmonic primes up to~$50\cdot 10^5$, finding a proportion close to the expected density. Our work extends previous computations by Boyd by a factor of about $30$ and $50$, respectively.

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arXiv2024

Laplace convolutions of weighted averages of arithmetical functions

Let $G(g;x):=\sum_{n\leq x}g(n)$ be the summatory function of an arithmetical function $g(n)$. In this paper, we prove that we can write weighted averages of an arbitrary fixed number $N$ of arithmetical functions $g_{j}(n),\,j\in\left\{ 1,\dots,N\right\} $ as an integral involving the convolution (in the sense of Laplace) of $G_{j}(x),\,j\in\left\{ 1,\dots,N\right\} $. Furthermore, we prove an identity that allows us to obtain known results about averages of arithmetical functions in a very simple and natural way, and overcome some technical limitations for some well-known problems.

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arXiv2023

$p$-adic valuation of harmonic sums and their connections with Wolstenholme primes

We explore a conjecture posed by Eswarathasan and Levine on the distribution of $p$-adic valuations of harmonic numbers $H(n)=1+1/2+\cdots+1/n$ that states that the set $J_p$ of the positive integers $n$ such that $p$ divides the numerator of $H(n)$ is finite. We proved two results, using a modular-arithmetic approach, one for non-Wolstenholme primes and the other for Wolstenholme primes, on an anomalous asymptotic behaviour of the $p$-adic valuation of $H(p^mn)$ when the $p$-adic valuation of $H(n)$ equals exactly 3.

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arXiv2020

A Cesàro average for an additive problem with an arbitrary number of prime powers and squares

In this paper we extend and improve all the previous results known in literature about weighted average, with Cesàro weight, of representations of an integer as sum of a positive arbitrary number of prime powers and a non-negative arbitrary number of squares. Our result includes all cases dealt with so far and allows us to obtain the best possible outcome using the chosen technique.

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arXiv2020

On the distribution of the digits of quotients of integers and primes

We investigate the distribution of the digits of quotients of randomly chosen positive integers taken from the interval $[1,T]$, improving the previously known error term for the counting function as $T\to+\infty$. We also resolve some natural variants of the problem concerning points with prime coordinates and points that are visible from the origin.

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arXiv2020

Identities for Catalan's constant arising from integrals depending on a parameter

In this paper we provide some relationships between Catalan's constant and the $_3{\rm F}_2$ and $_4{\rm F}_3$ hypergeometric functions, deriving them from some parametric integrals. In particular, using the complete elliptic integral of the first kind, we found an alternative proof of a result of Ramanujan for $_3{\rm F}_2$, a second identity related to $_4{\rm F}_3$ and using the complete elliptic integral of the second kind we obtain an identity by Adamchik.

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arXiv2019

On the average number of representations of an integer as a sum of like prime powers

We investigate the average number of representations of a positive integer as the sum of $k + 1$ perfect $k$-th powers of primes. We extend recent results of Languasco and the last Author, which dealt with the case $k = 2$ [6] and $k = 3$ [5] respectively. We use the same technique to study the corresponding problem for sums of just $k$ perfect $k$-th powers of primes.

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arXiv2019

A note on an average additive problem with prime numbers

We continue investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a short interval, whose admissible length depends on whether or not we assume the Riemann Hypothesis.

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arXiv2017

A Diophantine approximation problem with two primes and one $k$-th power of a prime

We refine a result of the last two Authors of [8] on a Diophantine approximation problem with two primes and a $k$-th power of a prime which was only proved to hold for $1<k<4/3$. We improve the $k$-range to $1<k\le 3$ by combining Harman's technique on the minor arc with a suitable estimate for the $L^4$-norm of the relevant exponential sum over primes $S_k$. In the common range we also give a stronger bound for the approximation.

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