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The sequence of fractional parts of roots

Kevin O'Bryant (2015)

Acta Arithmetica

We study the function M θ ( n ) = 1 / θ 1 / n , where θ is a positive real number, ⌊·⌋ and · are the floor and fractional part functions, respectively. Nathanson proved, among other properties of M θ , that if log θ is rational, then for all but finitely many positive integers n, M θ ( n ) = n / l o g θ - 1 / 2 . We extend this by showing that, without any condition on θ, all but a zero-density set of integers n satisfy M θ ( n ) = n / l o g θ - 1 / 2 . Using a metric result of Schmidt, we show that almost all θ have asymptotically (log θ log x)/12 exceptional n ≤ x. Using continued...

The set of minimal distances in Krull monoids

Alfred Geroldinger, Qinghai Zhong (2016)

Acta Arithmetica

Let H be a Krull monoid with class group G. Then every nonunit a ∈ H can be written as a finite product of atoms, say a = u 1 · . . . · u k . The set (a) of all possible factorization lengths k is called the set of lengths of a. If G is finite, then there is a constant M ∈ ℕ such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference d ∈ Δ*(H), where Δ*(H) denotes the set of minimal distances of H. We show that max Δ*(H) ≤ maxexp(G)-2,(G)-1 and that equality holds if every...

The terms of the form 7kx² in the generalized Lucas sequence with parameters P and Q

Olcay Karaatlı (2016)

Acta Arithmetica

Let Vₙ(P,Q) denote the generalized Lucas sequence with parameters P and Q. For all odd relatively prime values of P and Q such that P² + 4Q > 0, we determine all indices n such that Vₙ(P,Q) = 7kx² when k|P. As an application, we determine all indices n such that the equation Vₙ = 21x² has solutions.

Théorème des nombres premiers pour les fonctions digitales

Bruno Martin, Christian Mauduit, Joël Rivat (2014)

Acta Arithmetica

The aim of this work is to estimate exponential sums of the form n x Λ ( n ) e x p ( 2 i π ( f ( n ) + β n ) ) , where Λ denotes von Mangoldt’s function, f a digital function, and β ∈ ℝ a parameter. This result can be interpreted as a Prime Number Theorem for rotations (i.e. a Vinogradov type theorem) twisted by digital functions.

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