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Mathematics of Computation

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Algebraic numbers close to both 0 and $ 1$

Author: D. Zagier
Journal: Math. Comp. 61 (1993), 485-491
MSC: Primary 11R06; Secondary 11R04, 12D10
MathSciNet review: 1197513
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Abstract: A recent theorem of Zhang asserts that

$\displaystyle H(\alpha ) + H(1 - \alpha ) \geq C$

for all algebraic numbers $ \alpha \ne 0,1, (1 \pm \sqrt { - 3} )/2$, and some constant $ C > 0$. An elementary proof of this, with a sharp value for the constant, is given (the optimal value of C is $ \tfrac{1}{2}\log (\tfrac{1}{2}(1 + \sqrt 5 )) = 0,2406 \ldots $, attained for eight values of $ \alpha $) and generalizations to other curves are discussed.

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Article copyright: © Copyright 1993 American Mathematical Society