An asymptotic formula and some explicit estimates of the counting function of $y$-friable numbers

Saha, Biswajyoti; Sankaranarayanan, Ayyadurai; Suzuki, Yuta

doi:10.2969/aspm/08410367

VOL. 84 | 2020 An asymptotic formula and some explicit estimates of the counting function of $y$-friable numbers

Biswajyoti Saha, Ayyadurai Sankaranarayanan, Yuta Suzuki

Editor(s) Hidehiko Mishou, Takashi Nakamura, Masatoshi Suzuki, Yumiko Umegaki

Adv. Stud. Pure Math., 2020: 367-397 (2020) DOI: 10.2969/aspm/08410367

Abstract

We prove an asymptotic formula for the counting function of $y$-friable numbers in a certain range, using the saddle point method. Further, we also prove certain reasonable upper bounds for the counting function of $y$-friable numbers for various ranges of $y$ whose proofs are comparatively simpler than the earlier ones. Though these bounds need not be the best estimates, these are certainly useful in applications.