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Marco Cantarini

Publications and source records attributed to Marco Cantarini.

16 recordsLinked to original sources
arXiv2026

Averages of diagonal Elliott-Halberstam problem twisted by Möbius function with Sobolev and Hölder-Zygmund weights

Recalling that the so-called Elliott-Halberstam conjecture twisted by the Möbius function $μ(n)$ claims that \[ \sum_{q\leq N^θ}\max_{y\leq N}\max_{(a,q)=1}\left|\sum_{\underset{\scriptstyle n\equiv a\,\mod\,q}{n\leq y}}Λ(n)μ\left(N-n\right)-\frac{1}{φ\left(q\right)}\sum_{n\leq y}Λ(n)μ\left(N-n\right)\right|\ll\frac{N}{\log\left(N\right)^{A}} \] for every $A>0$, where $0<θ<1$ is fixed, and also recalling that the validity of this conjecture, in combination with the validity of the classical Elliott-Halberstam for suitable $θ$, proves the binary Goldbach conjecture, in this paper we study weighted average variants of this problem. We will show that, under Generalized Riemann Hypothesis, a weak version of the Gonek-Hejhal conjecture and working with weights belonging to the Sobolev space $W^{2,1}$ or in the Hölder-Zygmund spaces $\mathcal{C}^δ$ for suitable range of $δ$, the bound of the average is consistent with the bound of the ``diagonal versions'' of this conjecture (that is, taking $y=N$ and taking $n\equiv N\mod q)$. In particular, in the case of weights in Sobolev space, the consistent upper bound holds for the whole $0<θ<1$ and, in the case of weights in the Hölder-Zygmund class $\mathcal{C}^δ$, for $θ$ that depends on the choice of $δ$ but still not below the $1/2-2\varepsilon$ threshold.

math.NT↗
arXiv2026

On the discrete convolution of the Liouville and Möbius functions

In this article we study some properties of the discrete convolution of Liouville function $S(n):=\sum_{m_{1}+m_{2}=n}λ\left(m_{1}\right)λ\left(m_{2}\right)$, 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 $λ(n)$.

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arXiv2025

Wavefront Solutions for Reaction-diffusion-convection Models with Accumulation Term and Aggregative Movements

In this paper we analyze the wavefront solutions of parabolic partial differential equations of the type \[ g(u)u_τ+f(u)u_{x}=\left(D(u)u_{x}\right)_{x}+ρ(u),\quad u\left(τ,x\right)\in[0,1] \] where the reaction term $ρ$ is of monostable-type. We allow the diffusivity $D$ and the accumulation term $g$ to have a finite number of changes of sign. We provide an existence result of travelling wave solutions (t.w.s.) together with an estimate of the threshold wave speed. Finally, we classify the t.w.s. between classical and sharp ones.

math.AP↗
arXiv2024

Simultaneous approximation by neural network operators with applications to Voronovskaja formulas

In this paper, we considered the problem of the simultaneous approximation of a function and its derivatives by means of the well-known neural network (NN) operators activated by sigmoidal function. Other than a uniform convergence theorem for the derivatives of NN operators, we also provide a quantitative estimate for the order of approximation based on the modulus of continuity of the approximated derivative. Furthermore, a qualitative and quantitative Voronovskaja-type formula is established, which provides information about the high order of approximation that can be achieved by NN operators. To prove the above theorems, several auxiliary results involving sigmoidal functions have been established. At the end of the paper, noteworthy examples have been discussed in detail.

math.FA↗
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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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.

math.NT↗
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.

math.NT↗
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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arXiv2018

On the interplay between hypergeometric series, Fourier-Legendre expansions and Euler sums

In this work we continue the investigation about the interplay between hypergeometric functions and Fourier-Legendre ($\textrm{FL}$) series expansions. In the section "Hypergeometric series related to $π,π^2$ and the lemniscate constant", through the FL-expansion of $\left[x(1-x)\right]^μ$ (with $μ+1\in\frac{1}{4}\mathbb{N}$) we prove that all the hypergeometric series $$ \sum_{n\geq 0}\frac{(-1)^n(4n+1)}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^3,\quad \sum_{n\geq 0}\frac{(4n+1)}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^4,$$ $$\quad \sum_{n\geq 0}\frac{(4n+1)}{p(n)^2}\left[\frac{1}{4^n}\binom{2n}{n}\right]^4,\; \sum_{n\geq 0}\frac{1}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^3,\; \sum_{n\geq 0}\frac{1}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2 $$ return rational multiples of $\frac{1}π,\frac{1}{π^2}$ or the lemniscate constant, as soon as $p(x)$ is a polynomial fulfilling suitable symmetry constraints. Additionally, by computing the FL-expansions of $\frac{\log x}{\sqrt{x}}$ and related functions, we show that in many cases the hypergeometric $\phantom{}_{p+1} F_{p}(\ldots , z)$ function evaluated at $z=\pm 1$ can be converted into a combination of Euler sums. In particular we perform an explicit evaluation of $$ \sum_{n\geq 0}\frac{1}{(2n+1)^2}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2,\quad \sum_{n\geq 0}\frac{1}{(2n+1)^3}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2. $$ In the section "Twisted hypergeometric series" we show that the conversion of some $\phantom{}_{p+1} F_{p}(\ldots,\pm 1)$ values into combinations of Euler sums, driven by FL-expansions, applies equally well to some twisted hypergeometric series, i.e. series of the form $\sum_{n\geq 0} a_n b_n$ where $a_n$ is a Stirling number of the first kind and $\sum_{n\geq 0}b_n z^n = \phantom{}_{p+1} F_{p}(\ldots;z)$.

math.NT↗
arXiv2018

Some identities involving the Cesàro average of Goldbach numbers

Let $Λ\left(n\right)$ be the von Mangoldt function and $r_{G}\left(n\right) := \sum_{m_1 + m_2=n} Λ\left(m_1 \right) Λ\left(m_2 \right)$ be the counting function for the numbers that can be written as sum of two primes (that we will call "Goldbach numbers", for brevity) and let $\widetilde{S }\left(z\right) := \sum_{n\geq1} Λ\left(n\right) e^{-nz}$, with $z\in\mathbb{C}$, $\mathrm{Re}\left(z\right)>0$. In this paper we will prove the identity $$\widetilde{S}\left(z\right) = \frac{e^{-2z}}{z}-\sum_ρz^{-ρ} Γ\left(ρ\right) + \sum_ρ \left(z^{-ρ} γ\left(ρ,2z\right) - \frac{2^ρe^{-z}}ρ \right) + G\left(z\right)$$ where $γ\left(ρ,2z\right)$ is the lower incomplete Gamma function, $ρ=β+iγ$ runs over the non-trivial zeros of the Riemann Zeta function and $G\left(z\right)$ is a sum of (explicitly calculate) elementary function and complex Exponential integrals. In addition we will prove that \begin{align*} \sum_{n\leq N} r_G \left(n\right) \left(N-n\right) = & \frac{N^{3}}{6} - 2\sum_ρ\frac{\left(N-2\right)^{ρ+2}}{ρ\left(ρ+ 1\right)\left(ρ+2\right)} + & \sum_{ρ_1} \sum_{ρ_2} \frac{Γ\left(ρ_{1}\right) Γ\left(ρ_{2}\right)} {Γ\left(ρ_{1} + ρ_{2}+ 2\right)} N^{ρ_1 + ρ_2+1} + F\left(N\right) \end{align*} where $N>4$ is a natural number and $F\left(N\right)$ is a sum of (explicitly calculate) elementary functions, dilogarithms and sums over non-trivial zeros of the Riemann Zeta function involving the incomplete Beta function.

math.NT↗
arXiv2018

Explicit formula for the average of Goldbach and prime tuples representations

Let $Λ\left(n\right)$ be the Von Mangoldt function, let \[ r_{G}\left(n\right)=\underset{{\scriptstyle m_{1}+m_{2}=n}}{\sum_{m_{1},m_{2}\leq n}}Λ\left(m_{1}\right)Λ\left(m_{2}\right), \] \[ r_{PT}\left(N,h\right)=\sum_{n=0}^{N}Λ\left(n\right)Λ\left(n+h\right),\,h\in\mathbb{N} \] be the counting function of the Goldbach numbers and the counting function of the prime tuples, respectively. Let $N>2$ be an integer. We will find the explicit formulae for the averages of $r_{G}\left(n\right)$ and $r_{PT}\left(N,h\right)$ in terms of elementary functions, the incomplete Beta function $B_{z}\left(a,b\right)$, series over $ρ$ that, with or without subscript, runs over the non-trivial zeros of the Riemann Zeta function and the Dilogarithm function. We will also prove the explicit formulae in an asymptotic form and a truncated formula for the average of $r_{G}\left(n\right)$. Some observation about these formulae and the average with Cesàro weight \[ \frac{1}{Γ\left(k+1\right)}\sum_{n\leq N}r_{G}\left(n\right)\left(N-n\right)^{k},\,k>0 \] are included.

math.NT↗
arXiv2017

On the Cesàro average of the "Linnik numbers"

Let $Λ$ be the von Mangoldt function and $r_{Q}\left(n\right)=\sum_{m_{1}+m_{2}^{2}+m_{3}^{2}=n}Λ\left(m_{1}\right)$ be the counting function for the numbers that can be written as sum of a prime and two squares (that we will call Linnik numbers, for brevity). Let $N$ a sufficiently large integer, let $k>3/2$ and let $M_{i}\left(N,k\right),\, i=1,\dots,k$ suitable parameters depending on $J_{v}\left(u\right)$, where $J_{v}\left(u\right)$ denotes the Bessel function of complex order $v$ and real argument $u$. We prove that $$ \sum_{n\leq N}r_{Q}\left(n\right)\frac{\left(N-n\right)^{k}}{Γ\left(k+1\right)}=M_{1}\left(N,k\right)+M_{2}\left(N,k\right)+M_{3}\left(N,k\right)+M_{4}\left(N,k\right)+O\left(N^{k+1}\right).$$ We also prove that with this technique the bound $k>3/2$ is optimal.

math.NT↗
arXiv2017

On the Cesàro average of the numbers that can be written as sum of a prime and two squares of primes

Let $Λ\left(n\right)$ be the von Mangoldt function and $r_{SP}\left(n\right)=\sum_{m_{1}+m_{2}^{2}+m_{3}^{2}=n}Λ\left(m_{1}\right)Λ\left(m_{2}\right)Λ\left(m_{3}\right)$ be the counting function for the numbers that can be written as sum of a prime and two squares of primes. Let $N$ a sufficiently large integer, let $k>3/2$ and let $M_{i}\left(N,k\right),\, i=1,\dots,4$ suitable parameters depending on $Γ(s)$. We prove that $$\sum_{n\leq N}r_{SP}\left(n\right)\frac{\left(N-n\right)^{k}}{Γ\left(k+1\right)}=M_{1}\left(N,k\right)+M_{2}\left(N,k\right)+M_{3}\left(N,k\right)+M_{4}\left(N,k\right)+O\left(N^{k+1}\right).$$

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