A Dirichlet series is a series of the form $$F(s)=\sum_{n=1}^\infty {f(n)\over n^s},$$ where the variable $s$ may be complex or real and $f(n)$ is a number-theoretic function. The sum of the series, $F(s)$, is called the generating function of $f(n)$. The Riemann zeta-function $$\zeta(s)=\sum_{n=1}^{\infty}{1\over n^{s}}=\Pi_{p}\left(1-{1\over p^{s}}\right)^{-1},$$ where $n$ runs through all integers and $p$ runs through all primes is the special case where $f(n) =1$ identically. It is fundamental to the study of prime numbers and many generating functions are combinations of this function. In this paper, we give an overview of some of the commonly known number-theoretic functions together with their corresponding Dirichlet series.

Zykov proved that if $G$ and $\overline G$ are complementary graphs having chromatic numbers $\chi$ and $\overline \chi$, respectively then $\chi \cdot {\overline \chi}$ is at least the number of vertices of $G$. Nordhaus and Gaddum gave an upper bound for $\chi \cdot {\overline \chi}$ and gave both upper and lower bounds for the analogue $\chi + {\overline \chi}$. In this paper we characterize those graphs for which $\chi \cdot {\overline \chi}$ and $\chi+\overline{\chi}$ reach the bounds of Nordhaus and Gaddum.

Let $G$ be a finite group. Based on the prime graph of $G$, the order of $G$ can be divided into a product of co-prime positive integers. These integers are called order components of $G$ and the set of order components is denoted by $OC(G)$. Some non-abelian simple groups are known to be uniquely determined by their order components. In this paper we discuss the recognizability of simple $K_n$-groups ($n=3, 4$) by their order components.

Let $X$, $Y$, and $Z$ be random variables. If $X$ is positively correlated to $Y$ and $Y$ is positively correlated to $Z$, it does not necessarily follow that $X$ is positively correlated to $Z$. In this article we find ranges for the correlation coefficients $r_{XY}$ and $r_{YZ}$ that guarantee that $X$ and $Z$ have a specified level of correlation. We explore the implications of these results to finding relationships between variables investigated in different studies.

An exhaustive unconditional permutation distribution of a test statistic is necessary in the construction of exact tests of significance. Tables of exact critical values for the Mood test are scarce. In this paper, the exact permutation distribution of the Mood test statistic is generated empirically by actually obtaining all the distinct permutations of the variates in an experiment. The tables of exact critical values for the Mood test are produced.

A bijection between topological spaces, locally homeomorphic to the square of Sierpinski space, and graphs without isolated vertices is established. This is used to describe some topological properties of these spaces in combinatorial terms and to disprove a conjecture of Rostami [3] in particular.

We introduce and study the notion of sober partially ordered sets. Some questions about sober spaces are also stated.

We investigate a real-world problem about the measure of the intersection of periodic sets. The problem is modeled in discrete and continuous cases, and a solution is obtained from well-known theorems in undergraduate mathematics.