Explicit Estimates Involving the Primorial Integers and Applications
Safia Aoudjit and Djamel Berkane
University of Blida
Department of Mathematics
P. O. Box 270, Route de Soumaa
Blida
Algeria
Pierre Dusart
XLIM UMR 7252
Faculté des sciences et techniques
Université de Limoges
P. O. Box 123, avenue Albert Thomas
87060 Limoges Cedex
France
Abstract:
In this paper, we propose several explicit bounds for the function that
counts the number of primorial integers less than or equal to a given
positive real number. As applications, we obtain an effective version
of Pósa's inequality, and a method to estimate the maximal value
of the sum over prime divisors
Σp|q f(p) for a positive decreasing
function f, when q ranges over all integers less than
x. In particular,
we improve the log p upper bound for the maximal value of
Σp|q (log p)/(p—1).
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(Concerned with sequences
A000040
A002110
A056127
A111972
A233824.)
Received February 6 2021; revised versions received February 17 2021; February 19 2021; July 11 2021; July 28 2021.
Published in Journal of Integer Sequences,
July 28 2021.
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