Si $p_k$ est le k${}^{ème}$ nombre premier, $\theta \left(p_k\right)={\sum }_{i=1}^{k}log{p}_i$ la fonction de Chebyshev. Nous obtenons de nouvelles estimations et des améliorations des bornes données par Rosser et Schoenfeld, Schoenfeld et Robin pour les fonctions$$p_k,\theta \left(p_k\right),S_k=\sum _{i=1}^k{p}_i,\text{et}S\left(x\right)=\sum _{p\le x}p.$$Ces estimations sont obtenues en utilisant des méthodes basées sur l’intégrale de Stieltjes et par calcul direct pour les petites valeurs.

A positive integer $n$ is said to be a Jordan-Pólya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-Pólya numbers not exceeding a given number $x$.

Let $h$ be a complex valued multiplicative function. For any $N\in \mathbb{N}$, we compute the value of the determinant $D_N:={det}_{i|N,j|N}\left(\frac{h\left(\left(i,j\right)\right)}{ij}\right)$ where $\left(i,j\right)$ denotes the greatest common divisor of $i$ and $j$, which appear in increasing order in rows and columns. Precisely we prove that $$D_N=\prod _{p{}^l\parallel N}{\left(\frac{1}{p^{l\left(l+1\right)}}\stackrel{l}{\prod _{i=1}}\left(h\left(p^i\right)-h\left(p^{i-1}\right)\right)\right)}^{\tau \left(N/p^l\right)}.$$ This means that $D_N^{1/\tau \left(N\right)}$ is a multiplicative function of $N$. The algebraic apparatus associated with this result allows us to prove the following two results. The first one is the characterization of real multiplicative functions $f\left(n\right)$, with $0\le f\left(p\right)<1$, as minimal values of certain...

Nous caractérisons, dans cet article, les fonctions multiplicatives presque périodiques au sens de Bésicovith ayant un spectre de Fourier non vide.

Let α,β ∈ ℝ be fixed with α > 1, and suppose that α is irrational and of finite type. We show that there are infinitely many Carmichael numbers composed solely of primes from the non-homogeneous Beatty sequence ${ℬ}_{\alpha ,\beta }={(\lfloor \alpha n+\beta \rfloor )}_{n=1}^{\infty }$. We conjecture that the same result holds true when α is an irrational number of infinite type.

We consider the problem of determining whether a set of primes, or, more generally, prime ideals in a number field, can be realized as a finite union of residue classes, or of Frobenius conjugacy classes. We give necessary conditions for a set to be realized in this manner, and show that the subset of primes consisting of every other prime cannot be expressed in this way, even if we allow a finite number of exceptions.

The first author conjectured that Chebyshev-type prime bounds hold for Beurling generalized numbers provided that the counting function N(x) of the generalized integers satisfies the L¹ condition ${\int }_1^{\infty }|N\left(x\right)-Ax|dx/x^2<\infty$ for some positive constant A. This conjecture was shown false by an example of Kahane. Here we establish the Chebyshev bounds using the L¹ hypothesis and a second integral condition.