The Eratosthenes Progression p_{k+1} = π^{-1}(p_k), k=0,1,2,..., p_0 \in N \setminus P Determines An Inner Prime Number Distribution Law

Constellations of gaps in Eratosthenes sieve

A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences are known as constellations. Over the last year we identified a discrete linear system that exactly models the population of any gap across all stages of the sieve. In August 2014 we summarized our results from analyzing this discrete model on populations of single gaps. This paper extends the discrete system to model the populations of constellations of gaps…

Combinatorics of the gaps between primes

A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences are known as constellations. The populations of gaps and constellations across stages of Eratosthenes sieve are modeled exactly by discrete dynamic systems. These models and their asymptotic behaviors provide evidence on a number of open problems regarding gaps between prime numbers. For Eratosthenes sieve we show that the analogue of Polignac's conjecture is…

The first moment of primes in arithmetic progressions: Beyond the Siegel-Walfisz range

We investigate the first moment of primes in progressions $$ \sum_{\substack{q\leq x/N \\ (q,a)=1}} \Big(ψ(x; q, a) - \frac x{φ(q)}\Big) $$ as $x, N \to \infty$. We show unconditionally that, when $a=1$, there is a significant bias towards negative values, uniformly for $N\leq {\rm e}^{c\sqrt{\log x}}$. The proof combines recent results of the authors on the first moment and on the error term in the dispersion method. More generally, for $a \in \mathbb Z\setminus\{0\}$ we prove estimates that take into account the potential existence (or inexistence) of Landau-Siegel zeros.

Estimating constellations among primes - I. Uniformity

A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which are known as constellations. In this paper, we use those enumerations to estimate the numbers of these constellations that occur as constellations among prime numbers, and we compare these estimates with computational results. We include in our estimates the constellations corresponding to three and four consecutive primes in arithmetic progression. For these initial e…

Essentialité dans les bases additives

In this article we study the notion of essential subset of an additive basis, that is to say the minimal finite subsets $P$ of a basis $A$ such that $A \setminus P$ doesn't remains a basis. The existence of an essential subset for a basis is equivalent for this basis to be included, for almost all elements, in an arithmetic non-trivial progression. We show that for every basis $A$ there exists an arithmetic progression with a biggest common difference containing $A$. Having this common difference $a$ we are able to give an upper bound to the number of essential subsets of $A$: this is the radi…

算术级数中的素数丢番图逼近

Diophantine approximation with primes in an arithmetic progression

本文研究算术级数中素数对无理数的丢番图逼近，通过推广Vaughan定理得到逼近上界，并构造不可数集S使素数满足更强逼近，回答Erdős和Mahler的问题。