Linear independence of odd zeta values using Siegel's lemma
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866911290426916864 |
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| author | Fischler, Stéphane |
| author_facet | Fischler, Stéphane |
| contents | We prove that among 1 and the odd zeta values $ζ(3)$, $ζ(5)$, \ldots, $ζ(s)$, at least $ 0.21 \sqrt{s}/\sqrt{\log s}$ are linearly independent over the rationals, for any sufficiently large odd integer $s$. This is the first asymptotic improvement on the lower bound, logarithmic in $s$, obtained by Ball-Rivoal in 2001. The proof is based on Siegel's lemma to construct non-explicit linear forms in values at odd integers of the Riemann zeta function, instead of using explicit well-poised hypergeometric series. A new refinement of Siegel's linear independence criterion is applied, together with a multiplicity estimate (namely a generalization of Shidlovsky's lemma). The result is also adapted to deal with values of the first $s$ polylogarithms at a fixed algebraic point in the unit disk, improving bounds of Rivoal and Marcovecchio. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_10136 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Linear independence of odd zeta values using Siegel's lemma Fischler, Stéphane Number Theory 11J72 (Primary), 11M06 (Secondary) We prove that among 1 and the odd zeta values $ζ(3)$, $ζ(5)$, \ldots, $ζ(s)$, at least $ 0.21 \sqrt{s}/\sqrt{\log s}$ are linearly independent over the rationals, for any sufficiently large odd integer $s$. This is the first asymptotic improvement on the lower bound, logarithmic in $s$, obtained by Ball-Rivoal in 2001. The proof is based on Siegel's lemma to construct non-explicit linear forms in values at odd integers of the Riemann zeta function, instead of using explicit well-poised hypergeometric series. A new refinement of Siegel's linear independence criterion is applied, together with a multiplicity estimate (namely a generalization of Shidlovsky's lemma). The result is also adapted to deal with values of the first $s$ polylogarithms at a fixed algebraic point in the unit disk, improving bounds of Rivoal and Marcovecchio. |
| title | Linear independence of odd zeta values using Siegel's lemma |
| topic | Number Theory 11J72 (Primary), 11M06 (Secondary) |
| url | https://arxiv.org/abs/2109.10136 |