Articles written in Proceedings – Mathematical Sciences

    • The number of rationals determined by large sets of sifter integers


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      We prove that the number of fractions $h_1/h_2$ of integers $h_1$, $h_2$ a subset$\mathcal{A} \subset \mathcal{H}\cap [1, X]$ is at least $\alpha X/(\log X)^{3/2}$, where $\mathcal{H}$ is the set $p-1$, $p$ being a prime such that $p+1$ is a sum of two coprime squares. So, this number of fractions is $\gg _\epsilon \alpha^{1+\epsilon}\mid \mathcal{A}\mid^2$, where $\epsilon$ is any positive real number. We take this opportunity to describe a geometrical view of the sieve and its usage to study integer sequences.

    • A higher order Levin--Fa\u{i}nle\u{i}b theorem


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      When restricted to some non-negative multiplicative function, say $f$, boundedon primes and that vanishes on non square-free integers, our result provides us with an asymptotic for $\sum_{n\leq X} f (n)/n$ with error term $\mathcal{O}(({\rm log} X)^{\kappa - h - 1 + \epsilon}$ (for any positive $\epsilon > 0$) as soon as we have $\sum_{p\leq Q} f (p)(\log p)/p = \kappa \log Q +\eta +O(1/(\log 2Q)^h)$ for a non-negative $\kappa$ and some non-negative integer $h$. The method generalizes the 1967-approach of Levin and Fa\u{i}nle\u{i}b and uses a differential equation.

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