The parametrized family of metric Mahler measures
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arXiv
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| Format: | Preprint |
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2014
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| _version_ | 1866915221589721088 |
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| author | Samuels, Charles L. |
| author_facet | Samuels, Charles L. |
| contents | Let $M(α)$ denote the (logarithmic) Mahler measure of the algebraic number $α$. Dubickas and Smyth, and later Fili and the author, examined metric versions of $M$. The author generalized these constructions in order to associate, to each point in $t\in (0,\infty]$, a metric version $M_t$ of the Mahler measure, each having a triangle inequality of a different strength. We further examine the functions $M_t$, using them to present an equivalent form of Lehmer's conjecture. We show that the function $t\mapsto M_t(α)^t$ is constructed piecewise from certain sums of exponential functions. We pose a conjecture that, if true, enables us to graph $t\mapsto M_t(α)$ for rational $α$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1408_4883 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | The parametrized family of metric Mahler measures Samuels, Charles L. Number Theory 11R04, 11R09 (Primary), 30D20, 54A10 (Secondary) Let $M(α)$ denote the (logarithmic) Mahler measure of the algebraic number $α$. Dubickas and Smyth, and later Fili and the author, examined metric versions of $M$. The author generalized these constructions in order to associate, to each point in $t\in (0,\infty]$, a metric version $M_t$ of the Mahler measure, each having a triangle inequality of a different strength. We further examine the functions $M_t$, using them to present an equivalent form of Lehmer's conjecture. We show that the function $t\mapsto M_t(α)^t$ is constructed piecewise from certain sums of exponential functions. We pose a conjecture that, if true, enables us to graph $t\mapsto M_t(α)$ for rational $α$. |
| title | The parametrized family of metric Mahler measures |
| topic | Number Theory 11R04, 11R09 (Primary), 30D20, 54A10 (Secondary) |
| url | https://arxiv.org/abs/1408.4883 |