Smooth numbers and the Dickman ρ function-Reference-Cited by-同舟云学术

Smooth numbers and the Dickman ρ function

Published:2023-12 Issue:1 Volume:151 Page:139-169
ISSN:0021-7670
Container-title:Journal d'Analyse Mathématique
language:en
Short-container-title:JAMA

Author:

Gorodetsky Ofir

Abstract

AbstractWe establish an asymptotic formula for ψ(x, y) whose shape is xρ(log x/ log y) times correction factors. These factors take into account the contributions of zeta zeros and prime powers and the formula can be regarded as an (approximate) explicit formula for ψ(x, y). With this formula at hand we prove oscillation results for ψ(x, y), which resolve a question of Hildebrand on the range of validity of ψ(x, y) ≍ xρ(log x/ log y). We also address a question of Pomerance on the range of validity of ψ(x, y) ≥ xρ(log x/ log y).Along the way we improve classical estimates for ψ(x, y) and, on the Riemann Hypothesis, uncover an unexpected phase transition of ψ(x, y)at y = (log x)3/2+o(1).

Publisher

Springer Science and Business Media LLC

Subject

General Mathematics,Analysis

Link

https://link.springer.com/content/pdf/10.1007/s11854-023-0328-6.pdf

Reference27 articles.

1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. Government Printing Office, Washington, DC, 1964.

2. K. Alladi, The Turán–Kubilius inequality for integers without large prime factors, J. Reine Angew. Math. 335 (1982), 180–196.

3. K. Alladi, An Erdős–Kac theorem for integers without large prime factors, Acta Arith. 49 (1987), 81–105.

4. N. G. de Bruijn, The asymptotic behaviour of a function occurring in the theory of primes, J. Indian Math. Soc. (N.S.) 15 (1951), 25–32.

5. N. G. de Bruijn, On the number of positive integers ≤ x and free of prime factors ≥ y, Nederl. Acad. Wetensch. Proc. Ser. A. 54 (1951), 50–60.