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Alessandro Languasco

Publications and source records attributed to Alessandro Languasco.

At least 37 records · Page 2Linked to original sources
arXiv2019

A Cesàro Average of generalised Hardy-Littlewood numbers

We continue our recent work on additive problems with prime summands: we already studied the \emph{average} number of representations of an integer as a sum of two primes, and also considered individual integers. Furthermore, we dealt with representations of integers as sums of powers of prime numbers. In this paper, we study a Cesàro weighted partial \emph{explicit} formula for generalised Hardy-Littlewood numbers (integers that can be written as a sum of a prime power and a square) thus extending and improving our earlier results.

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arXiv2018

Short intervals asymptotic formulae for binary problems with prime powers, II

We improve some results about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell_1}+p_{2}^{\ell_2}$ and $n=p^{\ell_1} + m^{\ell_2}$, where $\ell_1, \ell_2\ge 2$ are fixed integers, $p,p_1,p_2$ are prime numbers and $m$ is an integer.

math.NT↗
arXiv2018

A Cesàro average for an additive problem with prime powers

In this paper we extend and improve our results on weighted averages for the number of representations of an integer as a sum of two powers of primes. Let $1\le \ell_1 \le \ell_2$ be two integers, $Λ$ be the von Mangoldt function and % \(r_{\ell_1,\ell_2}(n) = \sum_{m_1^{\ell_1} + m_2^{\ell_2}= n} Λ(m_1) Λ(m_2) \) % be the weighted counting function for the number of representation of an integer as a sum of two prime powers. Let $N \geq 2$ be an integer. We prove that the Cesàro average of weight $k > 1$ of $r_{\ell_1,\ell_2}$ over the interval $[1, N]$ has a development as a sum of terms depending explicitly on the zeros of the Riemann zeta-function.

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arXiv2018

Short intervals asymptotic formulae for binary problems with prime powers

We prove results about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell_1}+p_{2}^{\ell_2}$, with $\ell_1, \ell_2\in\{2,3\}$, $\ell_1+\ell_2\le 5$ are fixed integers, and $n=p^{\ell_1} + m^{\ell_2}$, with $\ell_1=2$ and $2\le \ell_2\le 11$ or $\ell_1=3$ and $ \ell_2=2$ are fixed integers, $p,p_1,p_2$ are prime numbers and $m$ is an integer.

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arXiv2018

Sums of four prime cubes in short intervals

We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}$, where $p_1,p_2,p_3,p_4$ are prime numbers, holds in intervals shorter than the the ones previously known.

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arXiv2018

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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arXiv2012

Explicit relations between primes in short intervals and exponential sums over primes

Under the assumption of the Riemann Hypothesis (RH), we prove explicit quantitative relations between hypothetical error terms in the asymptotic formulae for truncated mean-square average of exponential sums over primes and in the mean-square of primes in short intervals. We also remark that such relations are connected with a more precise form of Montgomery's pair-correlation conjecture.

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arXiv2012

On a ternary Diophantine problem with mixed powers of primes

Let $1 < k < 33 / 29$. We prove that if $λ_1$, $λ_2$ and $λ_3$ are non-zero real numbers, not all of the same sign and that $λ_1 / λ_2$ is irrational and $\varpi$ is any real number, then for any $\eps > 0$ the inequality $ \bigl|λ_1 p_1 + λ_2 p_2^2 + λ_3 p_3^k + \varpi \bigr| \le \bigl(\max_j p_j \bigr)^{-(33 - 29 k) / (72 k) + \eps} $ has infinitely many solutions in prime variables $p_1$, ..., $p_k$.

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arXiv2012

A Diophantine problem with prime variables

We study the distribution of the values of the form $λ_1 p_1 + λ_2 p_2 + λ_3 p_3^k$, where $λ_1$, $λ_2$ and $λ_3$ are non-zero real number not all of the same sign, with $λ_1 / λ_2$ irrational, and $p_1$, $p_2$ and $p_3$ are prime numbers. We prove that, when $1 < k < 4 / 3$, these value approximate rather closely any prescribed real number.

math.NT↗
arXiv2012

A Cesàro Average of Hardy-Littlewood numbers

Let $Λ$ be the von Mangoldt function and $r_{\textit{HL}}(n) = \sum_{m_1 + m_2^2 = n} Λ(m_1),$ be the counting function for the Hardy-Littlewood numbers. Let $N$ be a sufficiently large integer. We prove that $$\begin{align}\sum_{n \le N} r_{\textit{HL}}(n) \frac{(1 - n/N)^k}{Γ(k + 1)} &= \frac{π^{1 / 2}}2 \frac{N^{3 / 2}}{Γ(k + 5 / 2)} - \frac 12 \frac{N}{Γ(k + 2)} - \frac{π^{1 / 2}}2 \sum_ρ \frac{Γ(ρ)}{Γ(k + 3 / 2 + ρ)} N^{1 / 2 + ρ}\\ &+ 1/2 \sum_ρ \frac{Γ(ρ)}{Γ(k + 1 + ρ)} N^ρ + \frac{N^{3 / 4 - k / 2}}{π^{k + 1}} \sum_{\ell \ge 1} \frac{J_{k + 3 / 2} (2 π\ell N^{1 / 2})}{\ell^{k + 3 / 2}}\\ &- \frac{N^{1 / 4 - k / 2}}{π^k} \sum_ρ Γ(ρ) \frac{N^{ρ/ 2}}{π^ρ} \sum_{\ell \ge 1} \frac{J_{k + 1 / 2 + ρ} (2 π\ell N^{1 / 2})} {\ell^{k + 1 / 2 + ρ}} + \mathcal{O}_k(1).\end{align}$$ for $k > 1$, where $ρ$ runs over the non-trivial zeros of the Riemann zeta-function $ζ(s)$ and $J_ν (u)$ denotes the Bessel function of complex order $ν$ and real argument $u$.

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