On the representation of prime numbers in a given base.
Prof. Christian Mauduit
Mathematical Institute of Luminy, France
Let q be an integer greater or equal to 2.
In this talk, we will first give a survey on results concerning the
representation of prime numbers in base q. Then we will present a very recent
joint work with Joel Rivat answering a question by Alexandre Gelfond (1967)
concerning the sum of digits in base q function s.
In this work, we show that the expected proportion of prime numbers p is such
that s(p) belongs to a given arithmetic progression and that the sequences (s(p)x)
is uniformly distributed modulo 1 for any irrational number x.