On the number of partitions of a number into distinct divisors

Given a partition $\{E_0,\ldots,E_n\}$ of the set of primes and a vector $\mathbf{k} \in \mathbb{N}_0^{n+1}$, we compute an asymptotic formula for the quantity $|\{m \leq x: \omega_{E_j}(m) = k_j \ \forall \ 0 \leq j \leq n\}|$ uniformly in a wide range of the parameters $k_j$ that complements the results of a previous paper of the author. This is accomplished using an extension and generalization of a theorem of Wirsing due to the author that gives explicit estimates for the...

Let $Q(n)$ denote the count of the primitive subsets of the integers $\{1,2\ldots n\}$. We give a new proof that $Q(n) = \alpha^{(1+o(1))n}$ which allows us to give a good error term and to improve upon the lower bound for the value of this constant $\alpha$. We also show that the method developed can be applied to many similar problems that can be stated in terms of the divisor graph, including other questions about primitive sets, geometric-progression-free sets, and the di...

Let $\varphi(n)$ denote the Euler totient function. In this paper, we first establish a new upper bound for $n/\varphi(n)$ involving $K(n)$, the function that counts the number of primorials not exceeding $n$. In particular, this leads to an answer to a question raised by Aoudjit, Berkane, and Dusart concerning an upper bound for the sum-of-divisors function $\sigma(n)$. Furthermore, we give some lower bounds for $N_k/\varphi(N_k)$ as well as for $\sigma(N_k)/N_k$, where $N_k...

For $x\ge0$ let $\pi(x)$ be the number of primes not exceeding $x$. The asymptotic behaviors of the prime-counting function $\pi(x)$ and the $n$-th prime $p_n$ have been studied intensively in analytic number theory. Surprisingly, we find that $\pi(x)$ and $p_n$ have many combinatorial properties which should not be ignored. In this paper we pose 60 open problems on combinatorial properties of primes (including connections between primes and partition functions) for further r...

A classic theorem of Uchimura states that the difference between the sum of the smallest parts of the partitions of $n$ into an odd number of distinct parts and the corresponding sum for an even number of distinct parts is equal to the number of divisors of $n$. In this article, we initiate the study of the $k$th smallest part of a partition $\pi$ into distinct parts of any integer $n$, namely $s_k(\pi)$. Using $s_k(\pi)$, we generalize the above result for the $k$th smallest...

Given a positive integer $n$ the $k$-fold divisor function $d_k(n)$ equals the number of ordered $k$-tuples of positive integers whose product equals $n$. In this article we study the variance of sums of $d_k(n)$ in short intervals and establish asymptotic formulas for the variance of sums of $d_k(n)$ in short intervals of certain lengths for $k=3$ and for $k \ge 4$ under the assumption of the Lindel\"of hypothesis.

We show that if $N\pm 1=M\varphi(N)$ with $N\neq 15, 255$ composite, then $M<15.76515\log\log\log N$ and $M<16.03235\log\log\omega(N)$, together with similar results for the unitary totient function, Dedekind function, and the sum of unitary divisors.

For integers $0 < r \leq t$, let the function $D_{r,t}(n)$ denote the number of parts among all partitions of $n$ into distinct parts that are congruent to $r$ modulo $t$. We prove the asymptotic formula $$D_{r,t}(n) \sim \dfrac{3^{\frac 14} e^{\pi \sqrt{\frac{n}{3}}}}{2\pi t n^{\frac 14}} \left( \log(2) + \left( \dfrac{\sqrt{3} \log(2)}{8\pi} - \dfrac{\pi}{4\sqrt{3}} \left( r - \dfrac{t}{2} \right) \right) n^{- \frac 12} \right)$$ as $n \to \infty$. A corollary of this resul...

For $r \in \mathbb{R}, r> 1$ and $n \in \mathbb{Z}^+$, the divisor function $\sigma_{-r}$ is defined by $\sigma_{-r}(n) := \sum_{d \vert n} d^{-r}$. In this paper we show the number $C_r$ of connected components of $\overline{\sigma_{-r}(\mathbb{Z}^+)}$ satisfies $$\pi(r) + 1 \leq C_r \leq \frac{1}{2}\exp\left[\frac{1}{2}\dfrac{r^{20/9}}{(\log r)^{29/9}} \left( 1 + \frac{\log\log r}{\log r - \log\log r} + \frac{\mathcal{O}(1)}{\log r}\right) \right],$$ where $\pi(t)$ is the n...

In the paper we can prove that every integer can be written as the sum of two integers, one perfect square and one squarefree. We also establish the asympotic formula for the number of representations of an integer in this form. The result is deeply related with the divisor function. In the course of our study we get an independent result about it. Concretely we are able to deduce a new upper bound for the divisor function valid for any integer and fully explicit.