Prime Distribution and Siegel Zeroes
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915031816339456 |
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| author | Wright, Thomas |
| author_facet | Wright, Thomas |
| contents | Let $χ$ be a Dirichlet character mod $D$ with $L(s,χ)$ its associated $L$-function, and let $ψ(x,q,a)$ be, as usual, Chebyshev's prime-counting function for the primes of the arithmetic progression $a$ (mod $q$) with $(a,q)=1$. For a fixed $R>7$, we prove that under the assumption of an exceptional character $χ$ with $L(1,χ)<(\log D)^{-R}$, there exists a range of $x$ for which the asymptotic $$ψ(x,q,a)=\frac{ψ(x)}{ϕ(q)}\left(1-χ\left(\frac{aD}{(q,D)}\right)+o(1)\right)$$ holds for $q<x^{\frac{30}{59}-\varepsilon}$. We also show slightly better bounds for $q$ if we take an average over a range of $q$, finding an Elliott-Halberstam-type result for $q\sim Q$ on the range $Q<x^{\frac{16}{31}-\varepsilon}$. This improves on a Friedlander and Iwaniec 2003 result that requires $q<x^{\frac{233}{462}}$ and $R\geq 554,401^{554,401}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_12470 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Prime Distribution and Siegel Zeroes Wright, Thomas Number Theory Let $χ$ be a Dirichlet character mod $D$ with $L(s,χ)$ its associated $L$-function, and let $ψ(x,q,a)$ be, as usual, Chebyshev's prime-counting function for the primes of the arithmetic progression $a$ (mod $q$) with $(a,q)=1$. For a fixed $R>7$, we prove that under the assumption of an exceptional character $χ$ with $L(1,χ)<(\log D)^{-R}$, there exists a range of $x$ for which the asymptotic $$ψ(x,q,a)=\frac{ψ(x)}{ϕ(q)}\left(1-χ\left(\frac{aD}{(q,D)}\right)+o(1)\right)$$ holds for $q<x^{\frac{30}{59}-\varepsilon}$. We also show slightly better bounds for $q$ if we take an average over a range of $q$, finding an Elliott-Halberstam-type result for $q\sim Q$ on the range $Q<x^{\frac{16}{31}-\varepsilon}$. This improves on a Friedlander and Iwaniec 2003 result that requires $q<x^{\frac{233}{462}}$ and $R\geq 554,401^{554,401}$. |
| title | Prime Distribution and Siegel Zeroes |
| topic | Number Theory |
| url | https://arxiv.org/abs/2311.12470 |