Summary
LetR=R k andw=w k be the regulator and the number of roots of unity in the number fieldk. We determine allk for whichR/w<1/8. There are nine such fields. Sincew≥2, we have allk withR<1/4. In particular, we find the first three minima ofR over allk. There are three main ingredients to the proof:
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i)
A new analytic formula (Theorem A below).
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ii)
A refinement of Remak's geometric approach.
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iii)
A refinement of Zimmert's analytic method.
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Friedman, E. Analytic formulas for the regulator of a number field. Invent Math 98, 599–622 (1989). https://doi.org/10.1007/BF01393839
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DOI: https://doi.org/10.1007/BF01393839