arXiv subjects
Publications and source records attributed to Alessandro Languasco.
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 $ζ(s)$.
We refine a result of W.P. Li and Wang on the values of the form $ λ_1p_1 + λ_2p_2^{2} + λ_3p_3^{2} + μ_1 2^{m_1} +...+ μ_s 2^{m_s}, $ where $p_1,p_2,p_3$ are prime numbers, $m_1,..., m_s$ are positive integers, $λ_1,λ_2,λ_{3}$ are nonzero real numbers, not all of the same sign,$λ_2 / λ_3$ is irrational and $λ_i/μ_i \in \Q$, for $i\in\{1,2,3\}$.
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.