On an extension of a question of Baker. (English) Zbl 1520.11068
The purpose of this text is to prove linear independence results (over \(\mathbb Q\) or \(\overline{\mathbb Q}\)) concerning values at \(s=1\) of the \(L\)-functions \(L(s,\chi) = \sum_{n=1}^\infty \chi(n) n^{-s}\) associated with non-trivial Dirichlet characters \(\chi\) mod \(q\), or related numbers.
The main result reads as follows.
Theorem. Let \(q_1,\ldots,q_\ell \geq 3\) be pairwise coprime integers; assume that the products \(q_1\ldots q_\ell\) and \(\varphi(q_1)\ldots \varphi(q_\ell)\) are coprime, where \(\varphi\) is Euler’s totient function. Then all numbers \(L(1,\chi)\), where \(\chi\) is a non-trivial Dirichlet character modulo one of the \(q_j\)’s, are linearly independent over \(\mathbb Q\) (and even over the extension of \(\mathbb Q\) generated by a primitive \(\varphi(q_1)\ldots \varphi(q_\ell)\)-th root of unity).
When \(\ell=1\), this result was proved by A. Baker et al. [J. Number Theory 5, 224–236 (1973; Zbl 0267.10065)].
In the case \(\ell=1\), M. R. Murty and V. K. Murty have proved in [J. Number Theory 131, No. 9, 1723–1733 (2011; Zbl 1241.11083)] that if one restricts to even characters \(\chi\), it is possible to deduce linear independence over \(\overline{\mathbb Q}\) and drop the assumption \(\gcd(q_1,\varphi(q_1))=1\). In the paper under review, the authors prove this for any \(\ell\). Among other results, they also extend a result of Okada about linear independence of cotangent values over \(\mathbb Q\).
The main result reads as follows.
Theorem. Let \(q_1,\ldots,q_\ell \geq 3\) be pairwise coprime integers; assume that the products \(q_1\ldots q_\ell\) and \(\varphi(q_1)\ldots \varphi(q_\ell)\) are coprime, where \(\varphi\) is Euler’s totient function. Then all numbers \(L(1,\chi)\), where \(\chi\) is a non-trivial Dirichlet character modulo one of the \(q_j\)’s, are linearly independent over \(\mathbb Q\) (and even over the extension of \(\mathbb Q\) generated by a primitive \(\varphi(q_1)\ldots \varphi(q_\ell)\)-th root of unity).
When \(\ell=1\), this result was proved by A. Baker et al. [J. Number Theory 5, 224–236 (1973; Zbl 0267.10065)].
In the case \(\ell=1\), M. R. Murty and V. K. Murty have proved in [J. Number Theory 131, No. 9, 1723–1733 (2011; Zbl 1241.11083)] that if one restricts to even characters \(\chi\), it is possible to deduce linear independence over \(\overline{\mathbb Q}\) and drop the assumption \(\gcd(q_1,\varphi(q_1))=1\). In the paper under review, the authors prove this for any \(\ell\). Among other results, they also extend a result of Okada about linear independence of cotangent values over \(\mathbb Q\).
Reviewer: Stéphane Fischler (Paris)
MSC:
| 11J72 | Irrationality; linear independence over a field |
| 11J86 | Linear forms in logarithms; Baker’s method |
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 11M20 | Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\) |
Keywords:
Dirichlet \(L\) functions at \(1\); linear forms in logarithms; Ramachandra units; cotangent valuesReferences:
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