Primes in Arithmetic Progressions to Large Moduli and Siegel Zeroes
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| Format: | Preprint |
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2025
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| _version_ | 1866915490994061312 |
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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 Chebyshev's prime-counting function for primes congruent to $a$ modulo $q$. We show that under the assumption of an exceptional character $χ$ with $L(1,χ)=o\left((\log D)^{-5}\right)$, for any $q<x^{\frac 23-\varepsilon}$, the asymptotic $$ψ(x,q,a)=\frac{ψ(x)}{ϕ(q)}\left(1-χ\left(\frac{aD}{(D,q)}\right)+o(1)\right)$$ holds for almost all $a$ with $(a,q)=1$. We also find that for any fixed $a$, the above holds for almost all $q<x^{\frac 23-\varepsilon}$ with $(a,q)=1$. Previous prime equidistribution results under the assumption of Siegel zeroes (by Friedlander-Iwaniec and the current author) have found that the above asymptotic holds either for all $a$ and $q$ or on average over a range of $q$ (i.e. for the Elliott-Halberstam conjecture), but only under the assumption that $q<x^θ$ where $θ=\frac{30}{59}$ or $\frac{16}{31}$, respectively. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_10780 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Primes in Arithmetic Progressions to Large Moduli 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 Chebyshev's prime-counting function for primes congruent to $a$ modulo $q$. We show that under the assumption of an exceptional character $χ$ with $L(1,χ)=o\left((\log D)^{-5}\right)$, for any $q<x^{\frac 23-\varepsilon}$, the asymptotic $$ψ(x,q,a)=\frac{ψ(x)}{ϕ(q)}\left(1-χ\left(\frac{aD}{(D,q)}\right)+o(1)\right)$$ holds for almost all $a$ with $(a,q)=1$. We also find that for any fixed $a$, the above holds for almost all $q<x^{\frac 23-\varepsilon}$ with $(a,q)=1$. Previous prime equidistribution results under the assumption of Siegel zeroes (by Friedlander-Iwaniec and the current author) have found that the above asymptotic holds either for all $a$ and $q$ or on average over a range of $q$ (i.e. for the Elliott-Halberstam conjecture), but only under the assumption that $q<x^θ$ where $θ=\frac{30}{59}$ or $\frac{16}{31}$, respectively. |
| title | Primes in Arithmetic Progressions to Large Moduli and Siegel Zeroes |
| topic | Number Theory |
| url | https://arxiv.org/abs/2507.10780 |