A note on trigonometric polynomials for lower bounds of $ζ(s)$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912141292863488 |
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| author | Leong, Nicol Mossinghoff, Michael J. |
| author_facet | Leong, Nicol Mossinghoff, Michael J. |
| contents | Non-negative trigonometric polynomials satisfying certain properties are employed when studying a number of aspects of the Riemann zeta function. When establishing zero-free regions in the critical strip, the classical polynomial $3+4\cos(θ)+\cos(2θ)$ used by de la Vallée Poussin has since been replaced by more beneficial polynomials with larger degree. The classical polynomial was also employed by Titchmarsh to provide a lower bound on $|ζ(σ+it)|$ when $σ>1$. We show that this polynomial is optimal for this purpose. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05928 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on trigonometric polynomials for lower bounds of $ζ(s)$ Leong, Nicol Mossinghoff, Michael J. Number Theory Primary: 11M06, 42A05. Secondary: 11L03, 26D05 Non-negative trigonometric polynomials satisfying certain properties are employed when studying a number of aspects of the Riemann zeta function. When establishing zero-free regions in the critical strip, the classical polynomial $3+4\cos(θ)+\cos(2θ)$ used by de la Vallée Poussin has since been replaced by more beneficial polynomials with larger degree. The classical polynomial was also employed by Titchmarsh to provide a lower bound on $|ζ(σ+it)|$ when $σ>1$. We show that this polynomial is optimal for this purpose. |
| title | A note on trigonometric polynomials for lower bounds of $ζ(s)$ |
| topic | Number Theory Primary: 11M06, 42A05. Secondary: 11L03, 26D05 |
| url | https://arxiv.org/abs/2404.05928 |