On gaps between numbers with a large prime factor
The goal of this paper is to outline the proof of a conjecture of Gelfond [6] (1968) in a recent work in collaboration with Christian Mauduit [11] concerning the sum of digits of prime numbers, reflecting the lecture given in Edinburgh at the Journées Arithmétiques 2007.
Let and . Denote by the set of all integers whose canonical prime representation has all exponents
In 1989, E. Saias established an asymptotic formula for with a very good error term, valid for , , We extend this result to an algebraic number field by obtaining an asymptotic formula for the analogous function with the same error term and valid in the same region. Our main objective is to compare the formulae for and and in particular to compare the second term in the two expansions.
In this note, we study those positive integers which are divisible by , where is the Carmichael function.