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

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 $\zeta(s)$

Short intervals asymptotic formulae for binary problems with primes and powers, I: density 3/2

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

Short intervals asymptotic formulae for binary problems with primes and powers, II: density 1

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

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

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 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 a Diophantine problem with two primes and s powers of two

We refine a recent result of Parsell on the values of the form $\lambda_1p_1 + \lambda_2p_2 + \mu_1 2^{m_1} + ...m + \mu_s 2^{m_s},$ where $p_1,p_2$ are prime numbers, $m_1,...c, m_s$ are positive integers, $\lambda_1 / \lambda_2$ is negative and irrational and $\lambda_1 / \mu_1$, \lambda_2/\mu_2 \in \Q.Comment: v2: enlarged introduction, improved major arc estimat

A Diophantine approximation problem with two primes and one k-power of a prime

We refine a result of the last two Authors 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 643 by combining Harman's technique on the minor arc with a suitable estimate for the L4-norm of the relevant exponential sum over primes Sk. In the common range we also give a stronger bound for the approximation

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