Displaying similar documents to “On a conjecture of Mąkowski and Schinzel concerning the composition of the arithmetic functions σ and ϕ”

A new lower bound for ∥ ( 3 / 2 ) k ∥

Wadim Zudilin (2007)

Journal de Théorie des Nombres de Bordeaux

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We prove that, for all integers k exceeding some effectively computable number  K , the distance from ( 3 / 2 ) k to the nearest integer is greater than 0 . 5803 k .

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.