Related papers: Finite Differences of the Logarithm of the Partiti…

In the $1970$s Nicolas proved that the coefficients $p_d(n)$ defined by the generating function \begin{equation*} \sum_{n=0}^{\infty} p_d(n) \, q^n = \prod_{n=1}^{\infty} \left( 1- q^n\right)^{-n^{d-1}} \end{equation*} are log-concave for…

Sorting is a fundamental problem in computer science. In the classical setting, it is well-known that $(1\pm o(1)) n\log_2 n$ comparisons are both necessary and sufficient to sort a list of $n$ elements. In this paper, we study the Noisy…

Let $p_{-t}(n)$ denote the number of partitions of $n$ into $t$ colors. In analogy with Ramanujan's work on the partition function, Lin recently proved in \cite{Lin} that $p_{-3}(11n+7)\equiv0\pmod{11}$ for every integer $n$. Such…

We provide an elementary proof of the left side inequality and improve the right inequality in \bigg[\frac{n!}{x-(x^{-1/n}+\alpha)^{-n}}\bigg]^{\frac{1}{n+1}}&<((-1)^{n-1}\psi^{(n)})^{-1}(x)…

We prove that $$ \sum_{p \leq x} \frac{1}{\tau(p-1)} \asymp \frac{x}{(\log x)^{3/2}}, \quad \quad \sum_{n \leq x} \frac{1}{\tau(n^2+1)} \asymp \frac{x}{(\log x)^{1/2}}, $$ where $\tau(n)=\sum_{d|n}1$ is the number of divisors of $n$, and…

In 2016 Bessenrodt--Ono discovered an inequality addressing additive and multiplicative properties of the partition function. Generalization by several authors have been given; on partitions with rank in a given residue class by…

Define a "nuclear partition" to be an integer partition with no part equal to one. In this study we prove a simple formula to compute the partition function $p(n)$ by counting only the nuclear partitions of $n$, a vanishingly small subset…

We obtain estimates for the number $p_d(n)$ of $(d-1)$-dimensional integer partitions of a number $n$. It is known that the two-sided inequality $C_1(d)n^{1-1/d}<\log p_d(n)< C_2(d)n^{1-1/d}$ is always true and that $C_1(d)>1$ whenever…

The finite n-th polylogarithm li_n(z) in Z/p[z] is defined as the sum on k from 1 to p-1 of z^k/k^n. We state and prove the following theorem. Let Li_k:C_p to C_p be the p-adic polylogarithms defined by Coleman. Then a certain linear…

For a positive integer $n$, let $p(n)$ be the number of ways to express $n$ as a sum of positive integers. In this note, we revisit the derivation of the Rademacher's convergent series for $p(n)$ in a pedagogical way, with all the details…

Stanley defined a partition function t(n) as the number of partitions $\lambda$ of n such that the number of odd parts of $\lambda$ is congruent to the number of odd parts of the conjugate partition $\lambda'$ modulo 4. We show that t(n)…

We study a problem of Douglass and Ono concerning the smallest integer $n$ such that the partition function $p(n)$ begins with a specified string of digits $f$ in base $b$. By employing an elementary discrepancy framework, we establish new…

In this paper, we obtain upper and lower bounds for the partition function $p(n)$ by using an elementary geometric inequality in Euclidean space, and we extend the method to generalizations of the partition function.

Let $p_n$ denote the $n^{th}$ prime. Goldston, Pintz, and Yildirim recently proved that $ \liminf_{n\to \infty} \frac{(p_{n+1}-p_n)}{\log p_n} =0.$ We give an alternative proof of this result. We also prove some corresponding results for…

Given integer $n > 0$ and $m > 1$, we call a partition of set $[n] = \{1, \dots, n\}$ {\em $m$-good} if each of the partitioning sets is of size at most $m$ and the sum of numbers in it is a power of $m$, that is, $m^t$ for some $t \geq 0$.…

We study some arithmetic properties of the Ramanujan function $\tau(n)$, such as the largest prime divisor $P(\tau(n))$ and the number of distinct prime divisors $\omega(\tau(n))$ of $\tau(n)$ for various sequences of $n$. In particular, we…

The partition function, $p_A(n)$, is defined to be the number of partitions of $n$ with parts in the set A, where $n$ is a positive integer and $A$ is a set of positive integers. It is well documented that: if A is a finite set with…

Kurepa's hypothesis asserts that for each integer $n\ge 2$ the greatest common divisor of $!n:=\sum_{k=0}^{n-1}k!$ and $n!$ is $2$. Motivated by an equivalent formulation of this hypothesis involving derangement numbers, here we give a…

We visualize the identity p(n) = sum s(k) p(n-k)/n for the integer partition function p(n) involving the divisor function s, add comments on the history of visualizations of numbers, illustrate how different mathematical fields play…

Let $\bar{p}(n)$ denote the overpartition function. In this paper, we study the asymptotic higher order $\log$-concavity property of the overpatition function in a similar framework done by Hou and Zhang for the partition function. This…