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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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