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Transactions of the American Mathematical Society

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Sums of $k$ unit fractions


Author: Christian Elsholtz
Journal: Trans. Amer. Math. Soc. 353 (2001), 3209-3227
MSC (2000): Primary 11D68; Secondary 11D72, 11N36
DOI: https://doi.org/10.1090/S0002-9947-01-02782-9
Published electronically: April 12, 2001
MathSciNet review: 1828604
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Abstract | References | Similar Articles | Additional Information

Abstract: Erdős and Straus conjectured that for any positive integer $n\geq 2$ the equation $\frac {4}{n}= \frac {1}{x} + \frac {1}{y} + \frac {1}{z}$ has a solution in positive integers $x, y$, and $z$. Let $m > k \geq 3$ and \[ E_{m,k}(N)= \mid \{ n \leq N \mid \frac {m}{n} = \frac {1}{t_1} + \ldots + \frac {1}{t_k} \text { has no solution with }t_i \in \mathbb {N} \} \mid . \] We show that parametric solutions can be used to find upper bounds on $E_{m,k}(N)$ where the number of parameters increases exponentially with $k$. This enables us to prove \[ E_{m,k}(N) \ll N \exp \left ( -c_{m,k} (\log N)^{1-\frac {1}{2^{k-1}-1}} \right ) \text { with } c_{m,k}>0. \] This improves upon earlier work by Viola (1973) and Shen (1986), and is an “exponential generalization” of the work of Vaughan (1970), who considered the case $k=3$.


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Additional Information

Christian Elsholtz
Affiliation: Institut für Mathematik, Technische Universität Clausthal, Erzstrasse 1, D-38678 Clausthal-Zellerfeld, Germany
Email: elsholtz@math.tu-clausthal.de

Received by editor(s): May 23, 2000
Received by editor(s) in revised form: August 28, 2000
Published electronically: April 12, 2001
Additional Notes: The research for this paper was supported by a Ph.D. grant from the German National Merit Foundation
Article copyright: © Copyright 2001 American Mathematical Society