Primes in arithmetic progressions to large moduli and refinements of Harman's sieve
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914608231481344 |
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| author | Li, Runbo |
| author_facet | Li, Runbo |
| contents | We study the average distribution of primes of size $x$ in arithmetic progressions to moduli larger than $x^{\frac{1}{2}}$. Using arithmetic information from the works of many authors together with different variants of the original Harman's sieve, we construct suitable majorants and minorants for the prime indicator function $\mathbb{1}_{p}(n)$ that satisfy Bombieri--Vinogradov type mean value theorems with different types of moduli. Specifically, we obtain some mean value theorems for primes with bilinear forms of moduli up to $x^{\frac{9}{17}}$ or with trilinear forms of moduli up to $x^{\frac{17}{32}}$. As a by-product, we obtain new upper and lower bounds for $π(x; q, a)$ that hold for almost all moduli $q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20917 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Primes in arithmetic progressions to large moduli and refinements of Harman's sieve Li, Runbo Number Theory We study the average distribution of primes of size $x$ in arithmetic progressions to moduli larger than $x^{\frac{1}{2}}$. Using arithmetic information from the works of many authors together with different variants of the original Harman's sieve, we construct suitable majorants and minorants for the prime indicator function $\mathbb{1}_{p}(n)$ that satisfy Bombieri--Vinogradov type mean value theorems with different types of moduli. Specifically, we obtain some mean value theorems for primes with bilinear forms of moduli up to $x^{\frac{9}{17}}$ or with trilinear forms of moduli up to $x^{\frac{17}{32}}$. As a by-product, we obtain new upper and lower bounds for $π(x; q, a)$ that hold for almost all moduli $q$. |
| title | Primes in arithmetic progressions to large moduli and refinements of Harman's sieve |
| topic | Number Theory |
| url | https://arxiv.org/abs/2602.20917 |