Harmonik tipi toplamlar ve aritmetik özellikleri

The \emph{$n$-th harmonic number} is the sum of the first $n$ terms of the harmonic series, namely $$h_n = \sum_{k=1}^n \frac 1 k,$$ for $n \ge 1$. These numbers enjoy several arithmetic properties. For instance, it was first shown by Theisinger in 1915 that the $n$-th harmonic number is not an integer, when $n \ge 2.$ A generalization of these numbers was introduced by Conway and Guy in 1996. They defined \emph{the $n$-th hyperharmonic number of order $r$} as $$h_n^{(r)} = \sum_{k=1}^n h_k^{(r-1)},$$ where $h_n^{(1)} = h_n$ and $n,r \in \mathbb Z^+$ such that $n \ge 1$ and $r \ge 2$. In 2007, it was conjectured by Mez\H o that there does not exist any hyperharmonic integer except 1. In the same paper, he also proved that $h_n^{(r)}$ is not an integer, for any integer $n>1$ and $r \le 3.$ In this thesis, we improve the known upper bounds for the order of non-integer hyperharmonic numbers. In particular, we show that for any $r \le 35\, 001$ and $n>1$, the corresponding hyperharmonic number is not an integer. Also, if $n$ is even or a prime power, or $r$ is odd, then the same property holds for $h_n^{(r)}$. Moreover, using the prime numbers in short intervals, we deduce that almost all $(n,r)$ tuples give us hyperharmonic numbers which are not integers. On the contrary, our final analysis leads to the existence of hyperharmonic integers. More precisely, for $r=64\cdot(2^\alpha - 1) +32$, the hyperharmonic number $h_{33}^{(r)}$ is integer for 153 different values of $\alpha \mod{748\, 440}$, where the smallest $r$ is equal to $64\cdot(2^{2659} - 1) +32$. This construction yields the infinitude of hyperharmonic integers, and refutes the conjecture of Mez\H o.