Primes in arithmetic progressions under the presence of Landau-Siegel zeroes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917096105967616 |
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| author | Sachpazis, Stelios |
| author_facet | Sachpazis, Stelios |
| contents | Let $x\geqslant 2$ and assume that $a$ and $q$ are coprime positive integers. As usual, $ψ(x;q,a):=\sum_{n\leqslant x,n\equiv a(\!\!\!\mod{\!\!q})}Λ(n)$, where $Λ$ is the von Mangoldt function. In 2003, Friedlander and Iwaniec assumed the existence of exceptional characters corresponding to "extreme" Landau-Siegel zeroes and established a meaningful asymptotic formula for $ψ(x;q,a)$ beyond the square-root barrier of the Generalized Riemann Hypothesis. In particular, their asymptotic yields non-trivial information for moduli $q\leqslant x^{1/2+1/231}$. In this paper, we considerably relax the extremity of the Landau-Siegel zero required in the work of Friedlander and Iwaniec and obtain a conditional asymptotic formula for $ψ(x;q,a)$ in a slightly wider range of $q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_16452 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Primes in arithmetic progressions under the presence of Landau-Siegel zeroes Sachpazis, Stelios Number Theory 11N13, 11N36, 11N37 Let $x\geqslant 2$ and assume that $a$ and $q$ are coprime positive integers. As usual, $ψ(x;q,a):=\sum_{n\leqslant x,n\equiv a(\!\!\!\mod{\!\!q})}Λ(n)$, where $Λ$ is the von Mangoldt function. In 2003, Friedlander and Iwaniec assumed the existence of exceptional characters corresponding to "extreme" Landau-Siegel zeroes and established a meaningful asymptotic formula for $ψ(x;q,a)$ beyond the square-root barrier of the Generalized Riemann Hypothesis. In particular, their asymptotic yields non-trivial information for moduli $q\leqslant x^{1/2+1/231}$. In this paper, we considerably relax the extremity of the Landau-Siegel zero required in the work of Friedlander and Iwaniec and obtain a conditional asymptotic formula for $ψ(x;q,a)$ in a slightly wider range of $q$. |
| title | Primes in arithmetic progressions under the presence of Landau-Siegel zeroes |
| topic | Number Theory 11N13, 11N36, 11N37 |
| url | https://arxiv.org/abs/2511.16452 |