On primes in arithmetic progressions and bounded gaps between many primes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866915165487759360 |
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| author | Stadlmann, Julia |
| author_facet | Stadlmann, Julia |
| contents | We prove that the primes below $x$ are, on average, equidistributed in arithmetic progressions to smooth moduli of size up to $x^{1/2+1/40-ε}$. The exponent of distribution $\tfrac{1}{2} + \tfrac{1}{40}$ improves on a result of Polymath, who had previously obtained the exponent $\tfrac{1}{2} + \tfrac{7}{300}$. As a consequence, we improve results on intervals of bounded length which contain many primes, showing that $\liminf_{n \rightarrow \infty} (p_{n+m}-p_n) = O(\exp(3.8075 m))$. The main new ingredient of our proof is a modification of the q-van der Corput process. It allows us to exploit additional averaging for the exponential sums which appear in the Type I estimates of Polymath. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_00425 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On primes in arithmetic progressions and bounded gaps between many primes Stadlmann, Julia Number Theory 11N05, 11N36, 11L07 We prove that the primes below $x$ are, on average, equidistributed in arithmetic progressions to smooth moduli of size up to $x^{1/2+1/40-ε}$. The exponent of distribution $\tfrac{1}{2} + \tfrac{1}{40}$ improves on a result of Polymath, who had previously obtained the exponent $\tfrac{1}{2} + \tfrac{7}{300}$. As a consequence, we improve results on intervals of bounded length which contain many primes, showing that $\liminf_{n \rightarrow \infty} (p_{n+m}-p_n) = O(\exp(3.8075 m))$. The main new ingredient of our proof is a modification of the q-van der Corput process. It allows us to exploit additional averaging for the exponential sums which appear in the Type I estimates of Polymath. |
| title | On primes in arithmetic progressions and bounded gaps between many primes |
| topic | Number Theory 11N05, 11N36, 11L07 |
| url | https://arxiv.org/abs/2309.00425 |