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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.06109 |
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| _version_ | 1866914143881134080 |
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| author | Ray, Swapnil |
| author_facet | Ray, Swapnil |
| contents | In this report, we present a proof of Levinson's theorem, following the ideas of Matthew P. Young in 2010, which states that one-third of the non-trivial zeros of the Riemann zeta function lie on the critical line, i.e. the line Re(s) = 1/2, using a mollified second moment of the zeta-function. Later, we present a generalized result for Dirichlet L-functions by Xiaosheng Wu in 2018, using Levinson's method, showing that more than two-fifths of the non-trivial zeros of Dirichlet L-functions are on the critical line. Moreover, more than two-fifths of the non-trivial zeros are simple and on the critical line, using a longer mollifier than in Levinson's original proof. This generalizes a result by Conrey from 1989 that the Riemann zeta-function has at least two-fifths of its zeros on the critical line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06109 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Levinson's theorem and its generalization for Dirichlet L-functions Ray, Swapnil Number Theory In this report, we present a proof of Levinson's theorem, following the ideas of Matthew P. Young in 2010, which states that one-third of the non-trivial zeros of the Riemann zeta function lie on the critical line, i.e. the line Re(s) = 1/2, using a mollified second moment of the zeta-function. Later, we present a generalized result for Dirichlet L-functions by Xiaosheng Wu in 2018, using Levinson's method, showing that more than two-fifths of the non-trivial zeros of Dirichlet L-functions are on the critical line. Moreover, more than two-fifths of the non-trivial zeros are simple and on the critical line, using a longer mollifier than in Levinson's original proof. This generalizes a result by Conrey from 1989 that the Riemann zeta-function has at least two-fifths of its zeros on the critical line. |
| title | Levinson's theorem and its generalization for Dirichlet L-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2511.06109 |