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.Comment: Unconditional result improved by using a Robert-Sargos estimate (lemmas 6-7); more detailed proof of Lemma 5 inserted. Correction of a typo. 10 page

A Ces\`aro Average of Hardy-Littlewood numbers

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

The number of Goldbach representations of an integer

We prove the following result: Let $N \geq 2$ and assume the Riemann Hypothesis (RH) holds. Then $$\sum_{n=1}^{N} R(n) =\frac{N^{2}}{2} -2 \sum_{\rho} \frac{N^{\rho + 1}}{\rho (\rho + 1)} + O(N \log^{3}N),$$ where $\rho=1/2+i\gamma$ runs over the non-trivial zeros of the Riemann zeta function $\zeta(s)$

The Selberg integral and a new pair-correlation function for the zeros of the Riemann zeta-function

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.Comment: Proceedings of the Third Italian Meeting in Number Theory, Pisa, September 2015. To appear in the "Rivista di Matematica dell'Universita` di Parma

A Diophantine problem with prime variables

We study the distribution of the values of the form $\lambda_1 p_1 + \lambda_2 p_2 + \lambda_3 p_3^k$, where $\lambda_1$, $\lambda_2$ and $\lambda_3$ are non-zero real number not all of the same sign, with $\lambda_1 / \lambda_2$ 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.Comment: submitte

A Ces\`aro Average of Goldbach numbers

Let $\Lambda$ be the von Mangoldt function and $(r_G(n) = \sum_{m_1 + m_2 = n} \Lambda(m_1) \Lambda(m_2))$ be the counting function for the Goldbach numbers. Let $N \geq 2$ be an integer. We prove that $$\begin{align} &\sum_{n \le N} r_G(n) \frac{(1 - n/N)^k}{\Gamma(k + 1)} = \frac{N^2}{\Gamma(k + 3)} - 2 \sum_\rho \frac{\Gamma(\rho)}{\Gamma(\rho + k + 2)} N^{\rho+1}\\ &\qquad+ \sum_{\rho_1} \sum_{\rho_2} \frac{\Gamma(\rho_1) \Gamma(\rho_2)}{\Gamma(\rho_1 + \rho_2 + k + 1)} N^{\rho_1 + \rho_2} + \mathcal{O}_k(N^{1/2}), \end{align}$$ for $k > 1$, where $\rho$, with or without subscripts, runs over the non-trivial zeros of the Riemann zeta-function $\zeta(s)$.Comment: submitte

On the constant in the Mertens product for arithmetic progressions. I. Identities

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.Comment: References added, misprints corrected. 9 page

A Diophantine problem with a prime and three squares of primes

We prove that if $\lambda_1$, $\lambda_2$, $\lambda_3$ and $\lambda_4$ are non-zero real numbers, not all of the same sign, $\lambda_1 / \lambda_2$ is irrational, and $\varpi$ is any real number then, for any \eps > 0 the inequality \bigl|\lambda_1 p_1 + \lambda_2 p_2^2 + \lambda_3 p_3^2 + \lambda_4 p_4^2 + \varpi \bigr| \le \bigl(\max_j p_j \bigr)^{-1 / 18 + \eps} has infinitely many solution in prime variables $p_1$, ..., $p_4$Comment: submitte