Displaying similar documents to “Generalization of a theorem of Gauss-Kuzmin.”

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.

Special values of symmetric power L -functions and Hecke eigenvalues

Emmanuel Royer, Jie Wu (2007)

Journal de Théorie des Nombres de Bordeaux

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We compute the moments of L -functions of symmetric powers of modular forms at the edge of the critical strip, twisted by the central value of the L -functions of modular forms. We show that, in the case of even powers, it is equivalent to twist by the value at the edge of the critical strip of the symmetric square L -functions. We deduce information on the size of symmetric power L -functions at the edge of the critical strip in subfamilies. In a second part, we study the distribution of...

On a conjecture of Mąkowski and Schinzel concerning the composition of the arithmetic functions σ and ϕ

A. Grytczuk, F. Luca, M. Wójtowicz (2000)

Colloquium Mathematicae

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For any positive integer n let ϕ(n) and σ(n) be the Euler function of n and the sum of divisors of n, respectively. In [5], Mąkowski and Schinzel conjectured that the inequality σ(ϕ(n)) ≥ n/2 holds for all positive integers n. We show that the lower density of the set of positive integers satisfying the above inequality is at least 0.74.

Complete solutions of a family of cubic Thue equations

Alain Togbé (2006)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we use Baker’s method, based on linear forms of logarithms, to solve a family of Thue equations associated with a family of number fields of degree 3. We obtain all solutions to the Thue equation Φ n ( x , y ) = x 3 + ( n 8 + 2 n 6 - 3 n 5 + 3 n 4 - 4 n 3 + 5 n 2 - 3 n + 3 ) x 2 y - ( n 3 - 2 ) n 2 x y 2 - y 3 = ± 1 , for n ≥ 0 .