A new zero-density estimate for $ζ(s)$ and the error term in the Prime Number Theorem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911090076549120 |
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| author | Bellotti, Chiara |
| author_facet | Bellotti, Chiara |
| contents | We will provide a new type of zero-density estimate for $ζ(s)$ when $σ$ is sufficiently close to $1$. In particular, we will show that $N(σ,T)$ can be bounded by an absolute constant when $σ$ is sufficiently close to the left edge of the Korobov-Vinogradov zero-free region. As a consequence, we provide the optimal error term in the prime number theorem of the form $$ ψ(x)-x \ll x\exp \left\{-(1-\varepsilon) ω(x)\right\},\qquad ω(x):=\min _{t \geq 1}\{ν(t) \log x+\log t\}, $$ where $ν(t)=A_0(\log t)^{-2/3}(\log\log t)^{-1/3}$ is a decreasing function such that $ζ(σ+it)\neq 0$ for $σ\ge 1-ν(t)$. Precisely, we will show that we can take $\varepsilon=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02041 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new zero-density estimate for $ζ(s)$ and the error term in the Prime Number Theorem Bellotti, Chiara Number Theory 11M06, 11M26, 11N05, 11N56 We will provide a new type of zero-density estimate for $ζ(s)$ when $σ$ is sufficiently close to $1$. In particular, we will show that $N(σ,T)$ can be bounded by an absolute constant when $σ$ is sufficiently close to the left edge of the Korobov-Vinogradov zero-free region. As a consequence, we provide the optimal error term in the prime number theorem of the form $$ ψ(x)-x \ll x\exp \left\{-(1-\varepsilon) ω(x)\right\},\qquad ω(x):=\min _{t \geq 1}\{ν(t) \log x+\log t\}, $$ where $ν(t)=A_0(\log t)^{-2/3}(\log\log t)^{-1/3}$ is a decreasing function such that $ζ(σ+it)\neq 0$ for $σ\ge 1-ν(t)$. Precisely, we will show that we can take $\varepsilon=0$. |
| title | A new zero-density estimate for $ζ(s)$ and the error term in the Prime Number Theorem |
| topic | Number Theory 11M06, 11M26, 11N05, 11N56 |
| url | https://arxiv.org/abs/2508.02041 |