Sums of unit fractions

Abstract: Let $f_k(m,n)$ denote the number of solutions of $\frac{m}{n}= \frac{1}{x_1} + \cdots + \frac{1}{x_k}$ in positive integers $x_i$. The case $k=2$ is essentially a question on a divisor function, and the case $k=3$ is closely related to a sum of certain divisor functions. For the case $k=3, m=4$ Erd\H{o}s and Straus conjectured that \[ f_3(4,n)>0 \text{ for all } n>1. \] The case $m=n=1$ received special attention, and even has applications in discrete geometry. We give a survey on previous results and report on new results over the last years.

Theorem 1: There are infinitely many primes $p$ with \[ f_3(m,p)\gg\exp \left(c_m \frac{\log p}{\log \log p}\right). \]

Theorem 2: For fixed $m$ and almost all integers $n$ one has: \[ f_3(m,n)\gg (\log n)^{\log 3+o_m(1)}. \]

Theorem 3: $f_3(4,n)=O_{\varepsilon}\left(n^{3/5+\varepsilon}\right)$, for all $\varepsilon >0$. There are related but more complicated bounds when $k\geq 4$.

Joint work with T. Browning, S. Planitzer, and T. Tao.

number theory

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

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