|
ВИДЕОТЕКА |
“Numbers and functions” – Memorial conference for 80th birthday of Alexey Nikolaevich Parshin
|
|||
|
A construction of A. Schinzel — many numbers in a short interval without small prime factors S. V. Konyagin |
|||
Аннотация: Hardy and Littlewood (1923) conjectured that for any integers \begin{equation} \label{HL} \pi(x+y) \le \pi(x) + \pi(y). \end{equation} Let us call a set Let $$\max_{y\ge x}(\pi(x+y)-\pi(y))=\limsup_{y\ge x} (\pi(x+y)-\pi(y))=\rho^*(x).$$ Hensley and Richards (1974) proved that $$\rho^*(x) - \pi(x) \ge(\log 2- o(1)) x(\log x)^{-2}\quad(x\to\infty).$$ Therefore, (\ref{HL}) is not compatible with the prime $$\rho^*(x) - \pi(x) \ge((1/2)- o(1)) x(\log x)^{-2}\log\log\log x\quad(x\to\infty).$$ Язык доклада: английский |