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Can polylogarithms at algebraic points be linearly independent? (English) Zbl 1459.11144
For $$0\le x\leq 1$$ the $$s$$-th Lerch function is defined as $\Phi_s(x,z)=\sum_{k=0}^{\infty}\frac{z^{k+1}}{(k+x+1)^s}$ for $$s=1,2,\ldots,r$$. For $$x=0$$ this is the polylogarithmic function Li$$_s(z)$$.
For given pairwise distinct algebraic numbers $$\alpha_j$$ with $$0\leq |\alpha_j|\leq 1$$ $$(1\le j\le m)$$ the authors state a linear independence criterion over algebraic number fields of the numbers $$\Phi_i(x,\alpha_j)$$, for $$1\le s\le r$$, $$1\le j\le m$$ and 1. An explicit sufficient condition is given for the linear independence of values of the Lerch functions $$\Phi_i(x,z)$$, $$1\le s\le r$$, at $$m$$ distinct points in an algebraic number field of arbitrary finite degree without any assumptions on $$r$$ and $$m$$. For $$x=0$$, these results imply the linear independence of polylogarithms of distinct algebraic numbers of arbitrary degree, subject to a metric condition.
The proof bases on Padé approximation techniques.
The paper is illustrated by interesting examples. For $$|b|\ge e^{2715}$$ the numbers 1, Li$$_1(1/b)$$,\ldots,Li$$_{10}(1/b)$$,\ldots,Li$$_1(1/(10b))$$,\ldots,Li$$_{10}(1/(10b))$$ are linearly independent over $$\mathbb Q$$.

##### MSC:
 11G55 Polylogarithms and relations with $$K$$-theory 11J72 Irrationality; linear independence over a field 11J82 Measures of irrationality and of transcendence 11J86 Linear forms in logarithms; Baker’s method 11M35 Hurwitz and Lerch zeta functions 11D75 Diophantine inequalities 11D88 $$p$$-adic and power series fields
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