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Banach algebra techniques in the theory of arithmetic functions

Lutz G. Lucht (2008)

Acta Mathematica Universitatis Ostraviensis

For infinite discrete additive semigroups X [ 0 , ) we study normed algebras of arithmetic functions g : X endowed with the linear operations and the convolution. In particular, we investigate the problem of scaling the mean deviation of related multiplicative functions for X = log . This involves an extension of Banach algebras of arithmetic functions by introducing weight functions and proving a weighted inversion theorem of Wiener type in the frame of Gelfand’s theory of commutative Banach algebras.

Billiard complexity in the hypercube

Nicolas Bedaride, Pascal Hubert (2007)

Annales de l’institut Fourier

We consider the billiard map in the hypercube of d . We obtain a language by coding the billiard map by the faces of the hypercube. We investigate the complexity function of this language. We prove that n 3 d - 3 is the order of magnitude of the complexity.

Bornes effectives pour certaines fonctions concernant les nombres premiers

Jean-Pierre Massias, Guy Robin (1996)

Journal de théorie des nombres de Bordeaux

Si p k est le k è m e nombre premier, θ ( p k ) = i = 1 k log p i la fonction de Chebyshev. Nous obtenons de nouvelles estimations et des améliorations des bornes données par Rosser et Schoenfeld, Schoenfeld et Robin pour les fonctions p k , θ ( p k ) , S k = i = 1 k p i , et S ( x ) = p x p . Ces estimations sont obtenues en utilisant des méthodes basées sur l’intégrale de Stieltjes et par calcul direct pour les petites valeurs.

Bounds for the counting function of the Jordan-Pólya numbers

Jean-Marie De Koninck, Nicolas Doyon, A. Arthur Bonkli Razafindrasoanaivolala, William Verreault (2020)

Archivum Mathematicum

A positive integer n is said to be a Jordan-Pólya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-Pólya numbers not exceeding a given number x .

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