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The n -th prime asymptotically

Juan Arias de Reyna, Jérémy Toulisse (2013)

Journal de Théorie des Nombres de Bordeaux

A new derivation of the classic asymptotic expansion of the n -th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994).Realistic bounds for the error with li - 1 ( n ) , after having retained the first m terms, for 1 ≤ m ≤ 11 , are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible r 3 such that, for n ≥ r 3 , we have p n > s 3 ( n ) where s 3 ( n ) is the sum of the first four terms of the asymptotic expansion.

Théorème des nombres premiers pour les fonctions digitales

Bruno Martin, Christian Mauduit, Joël Rivat (2014)

Acta Arithmetica

The aim of this work is to estimate exponential sums of the form ∑ n ≤ x Λ ( n ) e x p ( 2 i π ( f ( n ) + β n ) ) , where Λ denotes von Mangoldt’s function, f a digital function, and β ∈ ℝ a parameter. This result can be interpreted as a Prime Number Theorem for rotations (i.e. a Vinogradov type theorem) twisted by digital functions.

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