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arXiv · 1803.08905

Many odd zeta values are irrational

Stéphane Fischler·Johannes Sprang·Wadim Zudilin
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Abstract

Building upon ideas of the second and third authors, we prove that at least $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ values of the Riemann zeta function at odd integers between 3 and $s$ are irrational, where $\varepsilon$ is any positive real number and $s$ is large enough in terms of $\varepsilon$. This lower bound is asymptotically larger than any power of $\log s$; it improves on the bound $\frac{1-\varepsilon}{1+\log2}\log s$ that follows from the Ball--Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.

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BibTeXRIS

Stéphane Fischler, Johannes Sprang, Wadim Zudilin. 2018-04-09. Many odd zeta values are irrational. https://doi.org/10.1112/s0010437x1900722x

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