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Documents  11A41 | enregistrements trouvés : 2

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Given a fixed integer $q \geq 2$, an irrational number $\xi$ is said to be a $q$-normal number if any preassigned sequence of $k$ digits occurs in the $q$-ary expansion of $\xi$ with the expected frequency, that is $1/q^k$. In this talk, we expose new methods that allow for the construction of large families of normal numbers. This is joint work with Professor Jean-Marie De Koninck.

11N37 ; 11K16 ; 11A41

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Multi angle  Prime numbers with preassigned digits
Swaenepoel, Cathy (Auteur de la Conférence) | CIRM (Editeur )

Bourgain (2015) estimated the number of prime numbers with a proportion $c$ > 0 of preassigned digits in base 2 ($c$ is an absolute constant not specified). We present a generalization of this result in any base $g$ ≥ 2 and we provide explicit admissible values for the proportion $c$ depending on $g$. Our proof, which adapts, develops and refines Bourgain’s strategy, is based on the circle method and combines techniques from harmonic analysis together with results on zeros of Dirichlet $L$-functions, notably a very sharp zero-free region due to Iwaniec.
Bourgain (2015) estimated the number of prime numbers with a proportion $c$ > 0 of preassigned digits in base 2 ($c$ is an absolute constant not specified). We present a generalization of this result in any base $g$ ≥ 2 and we provide explicit admissible values for the proportion $c$ depending on $g$. Our proof, which adapts, develops and refines Bourgain’s strategy, is based on the circle method and combines techniques from harmonic analysis ...

11N05 ; 11A41 ; 11A63

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