Abstract:
Hardy and Littlewood (1923) conjectured that for any integers
$x,y\ge2$ \begin{equation}
\label{HL}
\pi(x+y) \le \pi(x) + \pi(y).
\end{equation}
Let us call a set $\{b_1,\dots,b_k\}$ of integers admissible if for each
prime $p$ there is some congruence class $\bmod p$ which contains none
of the integers $b_i$. The prime $k$-tuple conjecture states that if a set
$\{b_1,\dots,b_k\}$ is admissible, then there exist infinitely many
integers $n$ for which all the numbers $n+b_1,\dots,n+b_k$ are primes.
Let $x$ be a positive integer and $\rho^*(x)$ be the maximum number
of integers in any interval $(y,y+x]$ (with no restriction on $y$)
which are relatively prime to all positive integers $\le x$.
The prime $k$-tuple conjecture implies that
$$\max_{y\ge x}(\pi(x+y)-\pi(y))=\limsup_{y\ge x} (\pi(x+y)-\pi(y))=\rho^*(x).$$
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 $k$-tuple
conjecture. Using a construction of Schinzel we show that
$$\rho^*(x) - \pi(x) \ge((1/2)- o(1)) x(\log x)^{-2}\log\log\log x\quad(x\to\infty).$$
Contact us: