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À la recherche de petites sommes d'exponentielles

Étienne Fouvry, Philippe Michel (2002)

Annales de l’institut Fourier

Soit f ( x ) une fraction rationnelle à coefficients entiers, vérifiant des hypothèses assez générales. On prouve l’existence d’une infinité d’entiers n , ayant exactement deux facteurs premiers, tels que la somme d’exponentielles x = 1 n exp ( 2 π i f ( x ) / n ) soit en O ( n 1 2 - β f ) , où β f > 0 est une constante ne dépendant que de la géométrie de f . On donne aussi des résultats de répartition du type Sato-Tate, pour certaines sommes de Salié, modulo n , avec n entier comme ci- dessus.

Gaps between primes in Beatty sequences

Roger C. Baker, Liangyi Zhao (2016)

Acta Arithmetica

We study the gaps between primes in Beatty sequences following the methods in the recent breakthrough by Maynard (2015).

On Linnik's theorem on Goldbach numbers in short intervals and related problems

Alessandro Languasco, Alberto Perelli (1994)

Annales de l'institut Fourier

Linnik proved, assuming the Riemann Hypothesis, that for any ϵ > 0 , the interval [ N , N + log 3 + ϵ N ] contains a number which is the sum of two primes, provided that N is sufficiently large. This has subsequently been improved to the same assertion being valid for the smaller gap C log 2 N , the added new ingredient being Selberg’s estimate for the mean-square of primes in short intervals. Here we give another proof of this sharper result which avoids the use of Selberg’s estimate and is therefore more in the spirit of Linnik’s...

Prime numbers along Rudin–Shapiro sequences

Christian Mauduit, Joël 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.

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