Ordinality and Riemann Hypothesis II
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909645352730624 |
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| author | Kim, Young Deuk |
| author_facet | Kim, Young Deuk |
| contents | For $\frac{1}{2}<x<1$, $y>0$, and $n\in\mathbb{N}$, let $\displaystyleθ_n(x+iy)=\sum_{i=1}^n\frac{\mbox{sgn}\, q_i}{q_i^{x+iy}}$, where $Q=\{q_1,q_2,q_3,\cdots\}$ is the set of finite products of distinct odd primes, and ${\mbox{sgn}}\, q=(-1)^k$ if $q$ is the product of $k$ distinct primes. In this paper, we prove that there exists an ordering of $Q$ such that the sequence $θ_n(x+iy)$ has a convergent subsequence. As an application, we study the Riemann hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07214 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ordinality and Riemann Hypothesis II Kim, Young Deuk Complex Variables Number Theory 11M26 For $\frac{1}{2}<x<1$, $y>0$, and $n\in\mathbb{N}$, let $\displaystyleθ_n(x+iy)=\sum_{i=1}^n\frac{\mbox{sgn}\, q_i}{q_i^{x+iy}}$, where $Q=\{q_1,q_2,q_3,\cdots\}$ is the set of finite products of distinct odd primes, and ${\mbox{sgn}}\, q=(-1)^k$ if $q$ is the product of $k$ distinct primes. In this paper, we prove that there exists an ordering of $Q$ such that the sequence $θ_n(x+iy)$ has a convergent subsequence. As an application, we study the Riemann hypothesis. |
| title | Ordinality and Riemann Hypothesis II |
| topic | Complex Variables Number Theory 11M26 |
| url | https://arxiv.org/abs/2401.07214 |