100% of the zeros of $ζ(s)$ are on the critical line
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866913405771710464 |
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| author | Suman, Shekhar |
| author_facet | Suman, Shekhar |
| contents | Throughout this manuscript the zeros are counted with multiplicity. We denote by $N(T)$ the number of zeros $ρ$ of $ζ(s)$ in the critical strip upto height $T$ where $T>3$ is not an ordinate of zero of $ζ(s)$. Denote by $N_0(T)$ the number of zeros $ρ$ of $ζ(s)$ on the critical line upto height $T$. We first show that there exists $ε_0>0$ such that $ξ(s)$ has no zeros on the boundary of a small rectangle $R_ε$ defined as $R_ε=\{σ+it\in\mathbb{C}\mid \frac{1}{2}-ε\leq σ\leq \frac{1}{2}+ε,\ 0\leq t\leq T\}$ whenever $0<ε<ε_0$. Secondly if $N_ε(T)$ is the number of zeros $ρ$ of $ζ(s)$ inside the rectangle $R_ε$ then we prove that $N_ε(T)=N_0(T)$ for $ε$ sufficiently small depending on the height $T$. We use the Littlewood's lemma on the rectangle $R_ε$ along with the Hadamard product of $ξ(s)$ and the asymptotic for the logarithmic derivative of $ζ(s)$ to prove that as $T\to \infty$, $$N_0(T)=\frac{T}{2π}\log\left(\frac{T}{2π}\right)-\frac{T}{2π}+\mathcal{O}(\log T)$$ Also if $κ$ is the proportion of zeros of $ζ(s)$ on the critical line $$κ:=\liminf_{T\to \infty} \frac{N_0(T)}{N(T)}$$ then we prove as a consequence that $κ=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_00811 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | 100% of the zeros of $ζ(s)$ are on the critical line Suman, Shekhar General Mathematics 11M26, 11M06 Throughout this manuscript the zeros are counted with multiplicity. We denote by $N(T)$ the number of zeros $ρ$ of $ζ(s)$ in the critical strip upto height $T$ where $T>3$ is not an ordinate of zero of $ζ(s)$. Denote by $N_0(T)$ the number of zeros $ρ$ of $ζ(s)$ on the critical line upto height $T$. We first show that there exists $ε_0>0$ such that $ξ(s)$ has no zeros on the boundary of a small rectangle $R_ε$ defined as $R_ε=\{σ+it\in\mathbb{C}\mid \frac{1}{2}-ε\leq σ\leq \frac{1}{2}+ε,\ 0\leq t\leq T\}$ whenever $0<ε<ε_0$. Secondly if $N_ε(T)$ is the number of zeros $ρ$ of $ζ(s)$ inside the rectangle $R_ε$ then we prove that $N_ε(T)=N_0(T)$ for $ε$ sufficiently small depending on the height $T$. We use the Littlewood's lemma on the rectangle $R_ε$ along with the Hadamard product of $ξ(s)$ and the asymptotic for the logarithmic derivative of $ζ(s)$ to prove that as $T\to \infty$, $$N_0(T)=\frac{T}{2π}\log\left(\frac{T}{2π}\right)-\frac{T}{2π}+\mathcal{O}(\log T)$$ Also if $κ$ is the proportion of zeros of $ζ(s)$ on the critical line $$κ:=\liminf_{T\to \infty} \frac{N_0(T)}{N(T)}$$ then we prove as a consequence that $κ=1$. |
| title | 100% of the zeros of $ζ(s)$ are on the critical line |
| topic | General Mathematics 11M26, 11M06 |
| url | https://arxiv.org/abs/2205.00811 |