Displaying similar documents to “On the distribution of the Euler function of shifted smooth numbers”

Smooth points of a semialgebraic set

Jacek Stasica (2003)

Annales Polonici Mathematici

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It is proved that the set of smooth points of a semialgebraic set is semialgebraic.

On the composition of the Euler function and the sum of divisors function

Jean-Marie De Koninck, Florian Luca (2007)

Colloquium Mathematicae

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Let H(n) = σ(ϕ(n))/ϕ(σ(n)), where ϕ(n) is Euler's function and σ(n) stands for the sum of the positive divisors of n. We obtain the maximal and minimal orders of H(n) as well as its average order, and we also prove two density theorems. In particular, we answer a question raised by Golomb.

Some finite generalizations of Euler's pentagonal number theorem

Ji-Cai Liu (2017)

Czechoslovak Mathematical Journal

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Euler's pentagonal number theorem was a spectacular achievement at the time of its discovery, and is still considered to be a beautiful result in number theory and combinatorics. In this paper, we obtain three new finite generalizations of Euler's pentagonal number theorem.

Asymptotics for a class of arithmetic functions

Wenguang Zhai (2015)

Acta Arithmetica

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We study the asymptotic behaviour of the summatory function of a class of arithmetic functions. These functions are generalizations of the well-known general 4-dimensional divisor function d₄(n). We show that the corresponding error estimate is the best one can obtain by the present methods of analytic number theory.