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Shidlovsky’s multiplicity estimate and irrationality of zeta values. (English) Zbl 1432.11084
Let \(\chi\) be a Dirichlet character modulo \(d\) of conductor \(N\). Let \(p\in \{ 0,1\}\) and \(a\geq 2\). Denote by \(\delta_{\chi ,p,a}\) the dimension of the \(\mathbb Q\)-vector vector space spanned by \(1\) and the numbers \(L(\chi ,s)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s}\) with \(2\leq s\leq a\) and \(s\equiv p \mod 2\). Then the author proves that \[ \delta_{\chi ,p,a}\geq \frac{1+o(1)}{N+\log 2}\log a. \] If in addition \(N\) is a multiple of \(4\) then we have \[ \delta_{\chi ,p,a}\geq \frac{1+o(1)}{\frac N2+\log 2}\log a. \] Here, \(o(1)\) is a sequence that depends on \(N\) and \(a\), and tends to \(0\) as \(a\to\infty\) for any \(N\).

MSC:
11J72 Irrationality; linear independence over a field
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
34M03 Linear ordinary differential equations and systems in the complex domain
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