arXiv subjects
Publications and source records attributed to A. Languasco.
In this paper we extend the well-known investigations of Montgomery and Goldston & Montgomery, concerning the pair-correlation function and its relations with the distribution of primes in short intervals, to a more general version of the pair-correlation function.
We refine a recent result of Parsell on the values of the form $λ_1p_1 + λ_2p_2 + μ_1 2^{m_1} + ...m + μ_s 2^{m_s}, $ where $p_1,p_2$ are prime numbers, $m_1,...c, m_s$ are positive integers, $λ_1 / λ_2$ is negative and irrational and $λ_1 / μ_1$, $λ_2/μ_2 \in \Q$.
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 $λ>1/2$ there exists a positive proportion of primes $p\leq X$ such that the interval $(p,p+ λ\log X]$ contains at least a prime number. As a consequence we improve Cheer and Goldston's result on the size of real numbers $λ>1$ with the property that there is a positive proportion of integers $m\leq X$ such that the interval $(m,m+ λ\log X]$ 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 $(m,m+ λ\log X]$ 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 $(p,p+ λ\log X]$ contains no primes.
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 $(a, q) = 1$.