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Leading digits and algebraic numbers. (Premiers chiffres significatifs et nombres algébriques.) (French. Abridged English version) Zbl 1004.11060
The aim of this note is to sketch a proof of the following result: there is a meromorphic continuation of \(\sum_{\{\log_\beta n\}<\log_\beta \alpha}n^{-s}\) to the whole plane if and only if \(\beta\) is a Pisot number, \(\alpha\in \mathbb{Q}(\beta)\), and either the second largest conjugate of \(\beta\) is real or the conjugate of \(\alpha\), corresponding to the second largest conjugate of \(\beta\) is positive. More precisely, the related sum \(\sum\{\log_\beta n/\alpha\}n^{-s}\) admits a meromorphic continuation to the whole plane if and only if \(\beta\) is a Pisot or a Salem number, and \(\alpha\in \mathbb{Q}(\beta)\); if \(\alpha\in{\mathcal M}_\beta:= \mathbb{Z}[1/\beta]/ f'(\beta)\) (\(f\) denotes the minimal polynomial of \(\beta\)), there is an analytic continuation if and only if the second largest conjugate of \(\beta\) is real; if \(\alpha\notin{\mathcal M}_\beta\), then for \(\beta\) a Pisot number there is a continuation if and only if there is an \(m\in \mathbb{Z}\) such that \(\text{trace} (m\alpha\beta^k)\) is an integer not divisible by \(m\) for all large \(k\).
This study is motivated by Duncan’s approach to Benford’s law [R. L. Duncan, Fibonacci Q. 7, 474-475 (1969; Zbl 0215.06706)] and by the lattice point problem under a logarithmic curve.
This note also contains a criterion for a polynomial of the form \(X^{n+1}+ aX^n+ bX+c\) to be the minimal polynomial of a Pisot number whose second largest conjugate is real.
MSC:
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11M41 Other Dirichlet series and zeta functions
11K50 Metric theory of continued fractions
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