On the exact location of the non-trivial zeros of Riemann's zeta function
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2013
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| _version_ | 1866911939302522880 |
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| author | de Reyna, Juan Arias van de Lune, Jan |
| author_facet | de Reyna, Juan Arias van de Lune, Jan |
| contents | In this paper we introduce the real valued real analytic function kappa(t) implicitly defined by exp(2 pi i kappa(t)) = -exp(-2 i theta(t)) * (zeta'(1/2-it)/zeta'(1/2+it)) and kappa(0)=-1/2. (where theta(t) is the function appearing in the known formula zeta(1/2+it)= Z(t) * e^{-i theta(t)}). By studying the equation kappa(t) = n (without making any unproved hypotheses), we will show that (and how) this function is closely related to the (exact) position of the zeros of Riemann's zeta(s) and zeta'(s). Assuming the Riemann hypothesis and the simplicity of the zeros of zeta(s), it will follow that the ordinate of the zero 1/2 + i gamma_n of zeta(s) will be the unique solution to the equation kappa(t) = n. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1305_3844 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | On the exact location of the non-trivial zeros of Riemann's zeta function de Reyna, Juan Arias van de Lune, Jan Number Theory 11M, 11M06, 11M26 In this paper we introduce the real valued real analytic function kappa(t) implicitly defined by exp(2 pi i kappa(t)) = -exp(-2 i theta(t)) * (zeta'(1/2-it)/zeta'(1/2+it)) and kappa(0)=-1/2. (where theta(t) is the function appearing in the known formula zeta(1/2+it)= Z(t) * e^{-i theta(t)}). By studying the equation kappa(t) = n (without making any unproved hypotheses), we will show that (and how) this function is closely related to the (exact) position of the zeros of Riemann's zeta(s) and zeta'(s). Assuming the Riemann hypothesis and the simplicity of the zeros of zeta(s), it will follow that the ordinate of the zero 1/2 + i gamma_n of zeta(s) will be the unique solution to the equation kappa(t) = n. |
| title | On the exact location of the non-trivial zeros of Riemann's zeta function |
| topic | Number Theory 11M, 11M06, 11M26 |
| url | https://arxiv.org/abs/1305.3844 |