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On the distribution of the partial sum of Euler's totient function in residue classes

100%
Lamzouri Y., Phaovibul M., Zaharescu A.
Colloquium Mathematicum
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2011
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tom 123
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nr 1
115-127
EN
We investigate the distribution of $Φ(n) = 1+ ∑_{i=1}ⁿ φ(i)$ (which counts the number of Farey fractions of order n) in residue classes. While numerical computations suggest that Φ(n) is equidistributed modulo q if q is odd, and is equidistributed modulo the odd residue classes modulo q when q is even, we prove that the set of integers n such that Φ(n) lies in these residue classes has a positive lower density when q = 3,4. We also provide a simple proof, based on the Selberg-Delange method, of a result of T. Dence and C. Pomerance on the distribution of φ(n) modulo 3.
2
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On consecutive integers divisible by the number of their divisors

100%
Andreescu T., Luca F., Phaovibul M.
Acta Arithmetica
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2016
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tom 173
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nr 3
269-281
EN
We prove that there are no strings of three consecutive integers each divisible by the number of its divisors, and we give an estimate for the number of positive integers n ≤ x such that each of n and n + 1 is a multiple of the number of its divisors.
3
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Diophantine approximation with partial sums of power series

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Berndt B., Kim S., Phaovibul M., Zaharescu A.
Acta Arithmetica
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2013
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tom 161
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nr 3
249-266
EN
We study the question: How often do partial sums of power series of functions coalesce with convergents of the (simple) continued fractions of the functions? Our theorems quantitatively demonstrate that the answer is: not very often. We conjecture that in most cases there are only a finite number of partial sums coinciding with convergents. In many of these cases, we offer exact numbers in our conjectures.
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