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Asymptotic nature of higher Mahler measure


Acta Arithmetica

We consider Akatsuka’s zeta Mahler measure as a generating function of the higher Mahler measure m k ( P ) of a polynomial P , where m k ( P ) is the integral of l o g k | P | over the complex unit circle. Restricting ourselves to P(x) = x - r with |r| = 1 we show some new asymptotic results regarding m k ( P ) , in particular | m k ( P ) | / k ! 1 / π as k → ∞.

Digits and continuants in euclidean algorithms. Ergodic versus tauberian theorems

Brigitte Vallée (2000)

Journal de théorie des nombres de Bordeaux

We obtain new results regarding the precise average-case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the average-case complexity of an algorithm is related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide...

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