Amplified Fourth Moment of the Riemann Zeta-Function and Applications
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| Format: | Preprint |
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2025
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| _version_ | 1866914163051200512 |
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| author | Bui, Hung M. Hall, Richard R. Jorge, Martin Subira |
| author_facet | Bui, Hung M. Hall, Richard R. Jorge, Martin Subira |
| contents | The twisted fourth moment of the Riemann zeta-function was established by Hughes and Young [J. Reine Angew. Math. 641 (2010), 203--236] and later improved by Bettin, Bui, Li and Radziwill [J. Eur. Math. Soc. (JEMS) 22 (2020), 3953--3980]. In applications one would often like to take the Dirichlet polynomial to mimic either $1/ζ^r(s)$ (a mollifier) or $ζ(s)^r$ (an amplifier) for some $r>0$. Previous known results include the mean value of the fourth power of $ζ(s)$ times the square or the fourth power of a mollifier, or the square of an amplifier. In this paper we obtain the asymptotic formula for the fourth moment of the Riemann zeta-function times the fourth power of an amplifier. This has various applications to the theory of the Riemann zeta-function, e.g. gaps between zeros of $ζ(s)$ and lower bounds for moments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_14415 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Amplified Fourth Moment of the Riemann Zeta-Function and Applications Bui, Hung M. Hall, Richard R. Jorge, Martin Subira Number Theory 11M26 The twisted fourth moment of the Riemann zeta-function was established by Hughes and Young [J. Reine Angew. Math. 641 (2010), 203--236] and later improved by Bettin, Bui, Li and Radziwill [J. Eur. Math. Soc. (JEMS) 22 (2020), 3953--3980]. In applications one would often like to take the Dirichlet polynomial to mimic either $1/ζ^r(s)$ (a mollifier) or $ζ(s)^r$ (an amplifier) for some $r>0$. Previous known results include the mean value of the fourth power of $ζ(s)$ times the square or the fourth power of a mollifier, or the square of an amplifier. In this paper we obtain the asymptotic formula for the fourth moment of the Riemann zeta-function times the fourth power of an amplifier. This has various applications to the theory of the Riemann zeta-function, e.g. gaps between zeros of $ζ(s)$ and lower bounds for moments. |
| title | Amplified Fourth Moment of the Riemann Zeta-Function and Applications |
| topic | Number Theory 11M26 |
| url | https://arxiv.org/abs/2511.14415 |