TY - JOUR

T1 - Smooth integers and de Bruijn's approximation λ

AU - Gorodetsky, Ofir

N1 - Publisher Copyright:
© 2023 The Author(s). Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

PY - 2023

Y1 - 2023

N2 - This paper is concerned with the relationship of -smooth integers and de Bruijn's approximation. Under the Riemann hypothesis, Saias proved that the count of -smooth integers up to, is asymptotic to when. We extend the range to 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. The term relates to the error term in the prime number theorem, and implies that large positive (resp. negative) values of lead to large positive (resp. negative) values of, and vice versa. Under the Linear Independence hypothesis, we show a Chebyshev's bias in.

AB - This paper is concerned with the relationship of -smooth integers and de Bruijn's approximation. Under the Riemann hypothesis, Saias proved that the count of -smooth integers up to, is asymptotic to when. We extend the range to 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. The term relates to the error term in the prime number theorem, and implies that large positive (resp. negative) values of lead to large positive (resp. negative) values of, and vice versa. Under the Linear Independence hypothesis, we show a Chebyshev's bias in.

KW - de Bruijn's approximation

KW - smooth integers

KW - smooth numbers

UR - http://www.scopus.com/inward/record.url?scp=85175996853&partnerID=8YFLogxK

U2 - 10.1017/prm.2023.115

DO - 10.1017/prm.2023.115

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AN - SCOPUS:85175996853

SN - 0308-2105

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

ER -