Joint distribution in residue classes of families of polynomially-defined additive functions
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929195928518656 |
|---|---|
| author | Roy, Akash Singha |
| author_facet | Roy, Akash Singha |
| contents | Let $g_1, \dots , g_M$ be additive functions for which there exist nonconstant polynomials $G_1, \dots , G_M$ satisfying $g_i(p) = G_i(p)$ for all primes $p$ and all $i \in \{1, \dots , M\}$. Under fairly general and nearly optimal hypotheses, we show that the functions $g_1, \dots , g_M$ are jointly equidistributed among the residue classes to moduli $q$ varying uniformly up to a fixed but arbitrary power of $\log x$. Thus, we obtain analogues of the Siegel-Walfisz Theorem for primes in arithmetic progressions, but with primes replaced by values of such additive functions. Our results partially extend work of Delange from fixed moduli to varying moduli, and also generalize recent work done for a single additive function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00892 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Joint distribution in residue classes of families of polynomially-defined additive functions Roy, Akash Singha Number Theory 2020: Primary 11A25, Secondary 11N36, 11N37, 11N64, 11N69 Let $g_1, \dots , g_M$ be additive functions for which there exist nonconstant polynomials $G_1, \dots , G_M$ satisfying $g_i(p) = G_i(p)$ for all primes $p$ and all $i \in \{1, \dots , M\}$. Under fairly general and nearly optimal hypotheses, we show that the functions $g_1, \dots , g_M$ are jointly equidistributed among the residue classes to moduli $q$ varying uniformly up to a fixed but arbitrary power of $\log x$. Thus, we obtain analogues of the Siegel-Walfisz Theorem for primes in arithmetic progressions, but with primes replaced by values of such additive functions. Our results partially extend work of Delange from fixed moduli to varying moduli, and also generalize recent work done for a single additive function. |
| title | Joint distribution in residue classes of families of polynomially-defined additive functions |
| topic | Number Theory 2020: Primary 11A25, Secondary 11N36, 11N37, 11N64, 11N69 |
| url | https://arxiv.org/abs/2401.00892 |