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On Gelfond’s conjecture about the sum of digits of prime numbers

Joël Rivat — 2009

Journal de Théorie des Nombres de Bordeaux

The goal of this paper is to outline the proof of a conjecture of Gelfond [] (1968) in a recent work in collaboration with Christian Mauduit [] concerning the sum of digits of prime numbers, reflecting the lecture given in Edinburgh at the Journées Arithmétiques 2007.

Prime numbers along Rudin–Shapiro sequences

Christian MauduitJoël Rivat — 2015

Journal of the European Mathematical Society

For a large class of digital functions f , we estimate the sums n x Λ ( n ) f ( n ) (and n x μ ( n ) f ( n ) , where Λ denotes the von Mangoldt function (and μ the Möbius function). We deduce from these estimates a Prime Number Theorem (and a Möbius randomness principle) for sequences of integers with digit properties including the Rudin-Shapiro sequence and some of its generalizations.

Fonctions digitales le long des nombres premiers

Bruno MartinChristian MauduitJoël Rivat — 2015

Acta Arithmetica

In a recent work we gave some estimations for exponential sums of the form n x Λ ( n ) e x p ( 2 i π ( f ( n ) + β n ) ) , where Λ denotes the von Mangoldt function, f a digital function, and β a real parameter. The aim of this work is to show how these results can be used to study the statistical properties of digital functions along prime numbers.

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