Sur les chiffres des nombres premiers translatés

Abstract

The aim of this work is to prove new results on a class of digital functions with special emphasis on shifted primes as arguments. Our method lies on the estimate of exponential sums of the form n x Λ(n) exp(2iπf (n + cn) + βn) where f a digital function, c = (cn) is an almostperiodic sequence in Z and β is a real parameter, which extend the works of and to the case of the shifted prime numbers satisfying a digital constraint.

Digital functions and sums involving the largest prime factor of an integer

“…Using elementary means, the authors of [13] improved (1.7) to the following (1.6). Then, there exists ρ f,b,q > 0 such that for every integer c ∈ Z and β ∈ R, j ∈ J 2 and x 2, we have…”

“…Later Mkaouar, Ouled Azaiez and Thuswaldner [13] improved these results to the case of the shifted prime numbers with a (T, θ)-almost periodic sequence satisfying a digital constraint (knowing that a sequence (c n ) is (T, θ)-almost periodic if for T ∈ N and 0 θ < 1 there exists a sequence (e n ) purely periodic such that the number of integers less than or equal to N satisfying c n ̸ = e n is O(N θ )). Their method lies on what follows Theorem C [13, Théorème 2.12].…”

“…The strongly q-additive functions have been extensively discussed in the literature, in particular their asymptotic distribution (see for instance [2,3,[11][12][13]). In order to extend the prime number theorem (originally proved by J. Hadamard and de la Vallée Poussin [6,9]) and theorem of I. M. Vinogradov on the equidistribution of {βp} if β ∈ R \ Q, and p runs over the primes [15] to the case of prime numbers verifying a digital constraint, the authors of [11] attacked this problem by studying the generalized exponential sum…”

Beatty sequences and prime numbers with restrictions on strongly q-additive functions

“…q-additive functions have been extensively discussed in the literature, in particular their asymptotic distribution (see for instance [4,6,8,[16][17][18][19][20]). Let F be the set of "digital functions" f = 0≤k<q a k |.| k , such that the real sequence a 0 , .…”

Strongly -Additive Functions and Distributional Properties of the Largest Prime Factor

“…Strongly q-additive functions, particularly their asymptotic distribution, have been extensively discussed in the literature (see, for example, [2,3,[10][11][12]). Let F be the set of digital functions f = 0≤k<q a k | • | k such that the real sequence a 0 , .…”