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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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