Smooth integers and de Bruijn's approximation Ʌ

Journal article

This paper is concerned with the relationship of 𝑦-smooth integers and de Bruijn's approximation Λ(𝑥, 𝑦). Under the Riemann hypothesis, Saias proved that the count of 𝑦-smooth integers up to 𝑥, Ψ(𝑥, 𝑦), is asymptotic to Λ(𝑥, 𝑦) when 𝑦 ≥ (log 𝑥)^2+𝜀. We extend the range to 𝑦 ≥ (log 𝑥)^3/2+𝜀 by introducing a correction factor that takes into account the contributions of zeta zeros and prime powers. We use this correction term to uncover a lower order term in the asymptotics of Ψ(𝑥, 𝑦) / Λ(𝑥, 𝑦). The term relates to the error term in the prime number theorem, and implies that large positive (resp. negative) values of ∑ _𝑛≤𝑦 Λ(𝑛) − 𝑦 lead to large positive (resp. negative) values of Ψ(𝑥, 𝑦) − Λ(𝑥, 𝑦), and vice versa. Under the Linear Independence hypothesis, we show a Chebyshev's bias in Ψ(𝑥, 𝑦) − Λ(𝑥, 𝑦).

Authors: Gorodetsky, O

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Cambridge University Press
• Journal: Proceedings of the Royal Society of Edinburgh Section A: Mathematics
• Volume: 155
• Issue: 3
• Pages: 792-820
• Publication date: 2023-10-31
• Acceptance date: 2023-10-05
• DOI: 10.1017/prm.2023.115
• EISSN: 1473-7124
• ISSN: 0308-2105
• Language: English
• Keywords: de Bruijn's approximation; smooth integers; smooth numbers