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

On the constant in the Mertens product for arithmetic progressions. II. Numerical values

We give explicit numerical values with 100 decimal digits for the constant in the Mertens product over primes in the arithmetic progressions $a \bmod q$, for $q \in \{3$, ..., $100\}$ and $(a, q) = 1$.Comment: The complete set of results can be downloaded from http://www.math.unipd.it/~languasc/MCcomput.html together with the source program in Gp and the results of the verifications of the consistency identities described in section

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

Prime numbers in logarithmic intervals

Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and h=\odi{X}. Then we will apply this to prove that for every $\lambda>1/2$ there exists a positive proportion of primes $p\leq X$ such that the interval $(p,p+ \lambda\log X]$ contains at least a prime number. As a consequence we improve Cheer and Goldston's result on the size of real numbers $\lambda>1$ with the property that there is a positive proportion of integers $m\leq X$ such that the interval $(m,m+ \lambda\log X]$ contains no primes. We also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers $m\leq X$ such that the interval $(m,m+ \lambda\log X]$ contains at least a prime number. The last application of these techniques are two theorems (the first one unconditional and the second one in which we assume the validity of the Riemann Hypothesis and of a form of the Montgomery pair correlation conjecture) on the positive proportion of primes $p\leq X$ such that the interval $(p,p+ \lambda\log X]$ contains no primes.Comment: 17 page

On the asymptotic formula for Goldbach numbers in short intervals

Let $R(k)=\sum\limits_{l+m=k}\Lambda(l)\Lambda(m)$, \Sing(k) = 2 \prod\limits_{p>2}\left(1-\frac{1}{(p-1)^2}\right) \prod\limits_{\substack{ p\mid k\\ p>2 }} \left(\frac{p-1}{p-2}\right) if $k$ is even and \Sing(k) =0 if $k$ is odd. It is known that R(k) \sim k\Sing(k) as $N\to \infty$ for almost all $k\in [N,2N]$ and that \sum_{k\in [n,n+H)}R(k) \sim \sum_{k\in [n,n+H)} k\Sing(k) \quad\hbox{for} \quad n\to \infty \eqno{(1)} uniformly for $H\geq n^{1/6+\epsilon}$. Here we prove, assuming $N^\epsilon\leq H\leq N^{1/6+\epsilon}$ and $N\to\infty$, that (1) holds for almost all $n\in [N,2N]$

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

References

Abstract of paper [1]. We study the distribution of the values of the form λ1p1 +λ2p2 +λ3p k 3, where λ1, λ2 and λ3 are non-zero real numbers not all of the same sign, with λ1/λ2 irrational, and p1, p2 and p3 are prime numbers. We prove that, when 1 < k < 4/3, these value approximate rather closely any prescribed real number