Sergei Konyagin: A construction of A. Schinzel – many numbers in a short interval without small prime factors

Hardy and Littlewood (1923) conjectured that for any integers x, y ≥ 2,

π(x + y) ≤ π(x) + π(y).    (1)

Let us call a set {b₁, . . . , b_k} of integers admissible if for each prime p there is some congruence class mod p which contains none of the integers b_i. The prime k-tuple conjecture states that if a set {b₁, . . . , b_k} is admissible, then there exist infinitely many integers n for which all the numbers n + b₁, . . . , n + b_k are primes.

Let x be a positive integer and ρ*(x) be the maximum number of integers in any interval (y, y +x] (with no restriction on y) that are relatively prime to all positive integers ≤ x. The prime k-tuple conjecture implies that

max_y≥x (π(x + y) − π(y)) = lim sup_y≥x (π(x + y) − π(y)) = ρ*(x).

Hensley and Richards (1974) proved that

ρ*(x) − π(x) ≥ (log 2 − o(1)) x (log x)⁻²    (x → ∞).

Therefore, (1) is not compatible with the prime k-tuple conjecture. Using a construction of Schinzel we show that

ρ*(x) − π(x) ≥ (log 2 − o(1)) x (log x)⁻² log log log x    (x → ∞).

This video is part of the Number Theory Web Seminar series.