On Littlewood's estimate for the modulus of the zeta function on the critical line
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914729709010944 |
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| author | Carneiro, Emanuel Milinovich, Micah B. |
| author_facet | Carneiro, Emanuel Milinovich, Micah B. |
| contents | Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_17803 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Littlewood's estimate for the modulus of the zeta function on the critical line Carneiro, Emanuel Milinovich, Micah B. Number Theory Classical Analysis and ODEs 11M06, 11M26, 41A30 Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms. |
| title | On Littlewood's estimate for the modulus of the zeta function on the critical line |
| topic | Number Theory Classical Analysis and ODEs 11M06, 11M26, 41A30 |
| url | https://arxiv.org/abs/2403.17803 |