arXiv subjects
Publications and source records attributed to Alessandro Zaccagnini.
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.
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.
We prove that a suitable explicit formula for the Cesaro-averaged number of representations of an integer as a sum of two primes holds in short intervals.
The present paper is a report on joint work with Alessandro Languasco and Alberto Perelli on our recent investigations on the Selberg integral and its connections to Montgomery's pair-correlation function. We introduce a more general form of the Selberg integral and connect it to a new pair-correlation function, emphasising its relations to the distribution of prime numbers in short intervals.
We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime square and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the unconditional case.
We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the unconditional case.
Assuming the Riemann Hypothesis we prove that the interval $[N, N + H]$ contains an integer which is a sum of a prime and two squares of primes provided that $H \ge C (\log N)^{4}$, where $C > 0$ is an effective constant.
We study an extension of Montgomery's pair-correlation conjecture and its relevance in some problems on the distribution of prime numbers.
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.
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 $p_1$, ..., $p_k$.
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.
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 $u$.
We prove that if $λ_1$, $λ_2$, $λ_3$ and $λ_4$ are non-zero real numbers, not all of the same sign, $λ_1 / λ_2$ 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_3^2 + λ_4 p_4^2 + \varpi \bigr| \le \bigl(\max_j p_j \bigr)^{-1 / 18 + \eps} $ has infinitely many solution in prime variables $p_1$, ..., $p_4$
Let $Λ$ be the von Mangoldt function and $(r_G(n) = \sum_{m_1 + m_2 = n} Λ(m_1) Λ(m_2))$ be the counting function for the Goldbach numbers. Let $N \geq 2$ be an integer. We prove that $$\begin{align} &\sum_{n \le N} r_G(n) \frac{(1 - n/N)^k}{Γ(k + 1)} = \frac{N^2}{Γ(k + 3)} - 2 \sum_ρ\frac{Γ(ρ)}{Γ(ρ+ k + 2)} N^{ρ+1}\\ &\qquad+ \sum_{ρ_1} \sum_{ρ_2} \frac{Γ(ρ_1) Γ(ρ_2)}{Γ(ρ_1 + ρ_2 + k + 1)} N^{ρ_1 + ρ_2} + \mathcal{O}_k(N^{1/2}), \end{align}$$ for $k > 1$, where $ρ$, with or without subscripts, runs over the non-trivial zeros of the Riemann zeta-function $ζ(s)$.
Assuming that the Generalized Riemann Hypothesis (GRH) holds, we prove an explicit formula for the number of representations of an integer as a sum of $k\geq 5$ primes. Our error terms in such a formula improve by some logarithmic factors an analogous result by Friedlander-Goldston.
We prove the following result: Let $N \geq 2$ and assume the Riemann Hypothesis (RH) holds. Then \[ \sum_{n=1}^{N} R(n) =\frac{N^{2}}{2} -2 \sum_ρ \frac{N^{ρ+ 1}}{ρ(ρ+ 1)} + O(N \log^{3}N), \] where $ρ=1/2+iγ$ runs over the non-trivial zeros of the Riemann zeta function $ζ(s)$.
We give explicit numerical values with 100 decimal digits for the Mertens constant involved in the asymptotic formula for $\sum\limits_{\substack{p\leq x p\equiv a \bmod{q}}}1/p$ and, as a by-product, for the Meissel-Mertens constant defined as $\sum_{p\equiv a \bmod{q}} (\log(1-1/p)+1/p)$, for $q \in \{3$, ..., $100\}$ and $(q, a) = 1$.
The aim of the paper is the proof of new identities for the constant in the Mertens product for arithmetic progressions. We deal with the problem of the numerical computation of these constants in another paper.