Displaying similar documents to “On the asymmetric divisor problem with congruence conditions”

Sign changes of error terms related to arithmetical functions

Paulo J. Almeida (2007)

Journal de Théorie des Nombres de Bordeaux

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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.

On the counting function for the generalized Niven numbers

Ryan Daileda, Jessica Jou, Robert Lemke-Oliver, Elizabeth Rossolimo, Enrique Treviño (2009)

Journal de Théorie des Nombres de Bordeaux

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Given an integer base q ≥ 2 and a completely q -additive arithmetic function f taking integer values, we deduce an asymptotic expression for the counting function N f ( x ) = # 0 ≤ n < x | f ( n ) ∣ n under a mild restriction on the values of f . When f = s q , the base q sum of digits function, the integers counted by N f are the so-called base q Niven numbers, and our result provides a generalization of the asymptotic known in that case.