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Periods of sets of lengths: a quantitative result and an associated inverse problem

Wolfgang A. Schmid (2008)

Colloquium Mathematicae

The investigation of quantitative aspects of non-unique factorizations in the ring of integers of an algebraic number field gives rise to combinatorial problems in the class group of this number field. In this paper we investigate the combinatorial problems related to the function 𝓟(H,𝓓,M)(x), counting elements whose sets of lengths have period 𝓓, for extreme choices of 𝓓. If the class group meets certain conditions, we obtain the value of an exponent in the asymptotic formula of this function...

Sign changes of error terms related to arithmetical functions

Paulo J. Almeida (2007)

Journal de Théorie des Nombres de Bordeaux

Let H ( x ) = ∑ n ≤ x φ ( n ) n - 6 π 2 x . Motivated by a conjecture of Erdös, Lau developed a new method and proved that # { n ≤ T : H ( n ) H ( n + 1 ) < 0 } ≫ T . We consider arithmetical functions f ( n ) = ∑ d ∣ n b d d whose summation can be expressed as ∑ n ≤ x f ( n ) = α x + P ( log ( x ) ) + E ( x ) , where P ( x ) is a polynomial, E ( x ) = - ∑ n ≤ y ( x ) b n n ψ x n + o ( 1 ) and ψ ( x ) = x - ⌊ x ⌋ - 1 / 2 . We generalize Lau’s method and prove results about the number of sign changes for these error terms.

Sur certaines équations fonctionnelles arithmétiques

Régis de La Bretèche, Gérald Tenenbaum (2000)

Annales de l'institut Fourier

Soit p k le k -ième nombre premier. Une fonction arithmétique complètement additive est définie sur ℕ * par la donnée des f ( p k ) et la formule f ( n ) = ∑ k ≥ 1 f ( p k ) v p k ( n ) ( n ≥ 1 ) , où v p désigne la...

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