Some results on the sums of unit fractions

Abstract

A unit fraction is a rational number having $1$ in its numerator and any positive integer in its denominator. This thesis is devoted to the investigation of various aspects of sums of unit fractions, a topic that covers a wide range of problems and techniques. An elementary example of such sums is the harmonic numbers. Given a positive integer $n,$ the $n^{th}$ harmonic number is defined as $$H_n = 1 + \frac{1}{2} + \dots + \frac{1}{n}.$$ We begin by presenting a generalization of the harmonic numbers called the Dedekind harmonic numbers. In order to define them, we take a number field $K$ and then consider the sum of reciprocals of norms of ideals of $\mathcal{O}_K$, the ring of integers of this number field $K$, whose norms are bounded by a given positive integer $n$. We first show that these numbers are not integers after a while. Then, we provide this specific upper bound for some quadratic number fields to guarantee that they are non-integer. Furthermore, under the Riemann hypothesis, we obtain the non-integerness of differences of these numbers together with uniform bounds for quadratic number fields and derive an asymptotic result. We then continue with another example of the sums of unit fractions called the hyperharmonic numbers. In his paper, Mez\H o proposed that these numbers are never integers, except for the trivial case $1$, and this conjecture remained unresolved for an extended period. Another question is also asked in the same paper: Can two hyperharmonic numbers of different indices and different orders be equal? A partial answer to a more generalized version of this question is given in this thesis, via a geometric approach with the help of related problems in arithmetic geometry. Afterwards, an analytic approach is followed and we deduce that the differences of distinct hyperharmonic numbers are almost never an integer. For any given prime number $p$, the set denoted by $J(p)$ was introduced by Eswarathasan and Levine. This set consists of the indices of the harmonic numbers whose numerators are divisible by this prime $p$ in their lowest terms. The size of this set for several prime numbers was calculated by several authors and some upper bounds for a counting function for this set were given. We generalize the set $J(p)$ to the generalized harmonic numbers. The generalized harmonic numbers are sums of unit fractions where they have some positive integer power $s$ of the positive integers in their denominators. We define the generalizations $J(p,s)$ and $J(p^s,s)$ of $J(p)$, deduce some finiteness results, provide congruence relations and eventually obtain an upper bound for the counting function for $J(p,s)$. Moreover, we provide an explicit criterion that implies the finiteness of our set, together with computational results, and then point out the subjects that may reveal more about the finiteness of $J(p,s)$, by introducing Bernoulli and Euler numbers together with the irregular primes.