Nombres premiers de la forme ⌊nc⌋

Abstract

For c > 1 we denote by πc(χ) the number of integers n ≤ χ such that ⌊nc⌋ is prime. In 1953, Piatetski-Shapiro has proved that π(χ) ∼ χ/c log χ, χ → +∞ holds for c < 12/11. Many authors have extended this range, which measures our progress in exponential sums techniques. In this article we obtain c < 1.16117. . . .