Linear independence of values of G-functions, II: outside the disk of convergence

Fischler, Stéphane; Rivoal, Tanguy

doi:10.1007/s40316-020-00135-5

Cited by 2 publications

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“…The usual version of this criterion (see for instance [9,Theorem 4]) is the same statement, but the assumption on F is replaced by the assumption that L (n) is invertible. The latter is stronger, since it is equivalent to asking F = R q for any n. Indeed if F = R q then L (n) has rank q: for each n we may extract q linearly independent columns of L (n) , and obtain an invertible matrix to which [9,Theorem 4] applies. The point is that we shall apply Proposition 2 to the matrices [s k,i ] constructed in Proposition 1, and the subspace F is not always equal to R q (see Remark 3 in §1.4).…”

Section: Siegel's Linear Independence Criterionmentioning

confidence: 99%

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Irrationality of values of L‐functions of Dirichlet characters

Fischler

2019

J. London Math. Soc.

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In a recent paper with Sprang and Zudilin, the following result was proved: if a is large enough in terms of ε > 0, then at least 2 (1−ε) log a log log a values of the Riemann zeta function at odd integers between 3 and a are irrational. This improves on the Ball-Rivoal theorem, that provides only 1−ε 1+log 2 log a such irrational values -but with a stronger property: they are linearly independent over the rationals.In the present paper we generalize this recent result to both L-functions of Dirichlet characters and Hurwitz zeta function. The strategy is different and less elementary: the construction is related to a Padé approximation problem, and a generalization of Shidlovsky's lemma is used to apply Siegel's linear independence criterion.We also improve the analogue of the Ball-Rivoal theorem in this setting: we obtain 1−ε 1+log 2 log a linearly independent values L(f, s) with s ≤ a of a fixed parity, when f is a Dirichlet character. The new point here is that the constant 1 + log 2 does not depend on f .

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Section: Siegel's Linear Independence Criterionmentioning

confidence: 99%

“…Using Eq. (2.2) we may apply the usual version of Siegel's criterion (namely [9,Theorem 4]) to this matrix and deduce that . . .…”

Section: Siegel's Linear Independence Criterionmentioning

confidence: 99%