Products of primes in arithmetic progressions
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866916125373104128 |
|---|---|
| author | Matomäki, Kaisa Teräväinen, Joni |
| author_facet | Matomäki, Kaisa Teräväinen, Joni |
| contents | A conjecture of Erdős states that, for any large prime $q$, every reduced residue class $\pmod q$ can be represented as a product $p_1p_2$ of two primes $p_1,p_2\leq q$. We establish a ternary version of this conjecture, showing that, for any sufficiently large cube-free integer $q$, every reduced residue class $\pmod q$ can be written as $p_1p_2p_3$ with $p_1,p_2,p_3\leq q$ primes. We also show that, for any $\varepsilon > 0$ and any sufficiently large integer $q$, at least $(2/3-\varepsilon)φ(q)$ reduced residue classes $\pmod q$ can be represented as a product $p_1 p_2$ of two primes $p_1, p_2 \leq q$. The problems naturally reduce to studying character sums. The main innovation in the paper is the establishment of a multiplicative dense model theorem for character sums over primes in the spirit of the transference principle. In order to deal with possible local obstructions we use bounds for the logarithmic density of primes in certain unions of cosets of subgroups of $\mathbb{Z}_q^\times$ of small index and study in detail the exceptional case that there exists a quadratic character $ψ\pmod{q}$ such that $ψ(p) = -1$ for almost all primes $p \leq q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_07679 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Products of primes in arithmetic progressions Matomäki, Kaisa Teräväinen, Joni Number Theory 11N13, 11B13, 11N36 A conjecture of Erdős states that, for any large prime $q$, every reduced residue class $\pmod q$ can be represented as a product $p_1p_2$ of two primes $p_1,p_2\leq q$. We establish a ternary version of this conjecture, showing that, for any sufficiently large cube-free integer $q$, every reduced residue class $\pmod q$ can be written as $p_1p_2p_3$ with $p_1,p_2,p_3\leq q$ primes. We also show that, for any $\varepsilon > 0$ and any sufficiently large integer $q$, at least $(2/3-\varepsilon)φ(q)$ reduced residue classes $\pmod q$ can be represented as a product $p_1 p_2$ of two primes $p_1, p_2 \leq q$. The problems naturally reduce to studying character sums. The main innovation in the paper is the establishment of a multiplicative dense model theorem for character sums over primes in the spirit of the transference principle. In order to deal with possible local obstructions we use bounds for the logarithmic density of primes in certain unions of cosets of subgroups of $\mathbb{Z}_q^\times$ of small index and study in detail the exceptional case that there exists a quadratic character $ψ\pmod{q}$ such that $ψ(p) = -1$ for almost all primes $p \leq q$. |
| title | Products of primes in arithmetic progressions |
| topic | Number Theory 11N13, 11B13, 11N36 |
| url | https://arxiv.org/abs/2301.07679 |