Pseudodifferential arithmetic, Riemann and Lindelöf hypotheses
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916062475321344 |
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| author | Unterberger, André |
| author_facet | Unterberger, André |
| contents | The Weyl symbolic calculus of operators leads to the construction, if one takes for symbol a certain distribution decomposing over the zeros of the Riemann zeta function, of an operator with the following property: the Riemann hypothesis is equivalent to the validity of a collection of estimates involving this operator. Pseudodifferential arithmetic, a novel chapter of pseudodifferential operator theory, makes it possible to make the operator under study fully explicit. This leads to a disproof of the conjecture: the closure of the set of real parts of non-trivial zeros of zeta is dense in $(0,1)$. A similar method leads to a proof of the Lindel\öf hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_12937 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Pseudodifferential arithmetic, Riemann and Lindelöf hypotheses Unterberger, André Number Theory 11M26, 35S99 The Weyl symbolic calculus of operators leads to the construction, if one takes for symbol a certain distribution decomposing over the zeros of the Riemann zeta function, of an operator with the following property: the Riemann hypothesis is equivalent to the validity of a collection of estimates involving this operator. Pseudodifferential arithmetic, a novel chapter of pseudodifferential operator theory, makes it possible to make the operator under study fully explicit. This leads to a disproof of the conjecture: the closure of the set of real parts of non-trivial zeros of zeta is dense in $(0,1)$. A similar method leads to a proof of the Lindel\öf hypothesis. |
| title | Pseudodifferential arithmetic, Riemann and Lindelöf hypotheses |
| topic | Number Theory 11M26, 35S99 |
| url | https://arxiv.org/abs/2208.12937 |