Estimates on the decay of the Laplace–Pólya integral - Ambrus Gergely (BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 2025)
The Laplace–Pólya integral, defined by , appears in several areas of mathematics. We study this quantity by combinatorial methods; accordingly, our investigation focuses on the values at integer . Our main result establishes a lower bound for the ratio which extends and generalises the previous estimates of Lesieur and Nicolas [23], and provides a natural counterpart to the upper estimate established in our previous work [2]. We derive the statement by purely combinatorial, elementary arguments. As a corollary, we deduce that no subdiagonal central sections of the unit cube are extremal, apart from the minimal, maximal, and the main diagonal sections. We also prove several consequences for Eulerian numbers.