Tanguy RIVOAL On the distribution of
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چکیده
Hawkins introduced a probabilistic version of Erathosthenes’ sieve and studied the associated sequence of random “primes” (pk)k≥1. Using various probabilistic techniques, many authors have obtained sharp results concerning these random “primes”, which are often in agreement with certain classical theorems or conjectures for prime numbers. In this paper, we prove that the number of integers k ≤ n such that pk+α−pk = α is almost surely equivalent to n/ log(n), for a given fixed integer α ≥ 1. This is a particular case of a recent result of Bui and Keating (differently formulated) but our method is different and enables us to provide an error term. We also prove that the number of integers k ≤ n such that pk ∈ aN+ b is almost surely equivalent to n/a, for given fixed integers a ≥ 1 and 0 ≤ b ≤ a − 1, which is an analogue of Dirichlet’s theorem.
منابع مشابه
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تاریخ انتشار 2009