In this paper, we present an integral inequality for the polar derivative of polynomials. Our theorem includes as special cases several interesting generalizations of some Zygmund type inequalities for polynomials.

The $abc$ conjecture, one of the most famous open problems in number theory, claims that three relatively prime positive integers $a,b,c$ satisfying $a+b=c$ cannot simultaneously have significant repetition among their prime factors; in particular, the product of the distinct primes dividing the three integers should never be much less than $c$. Triples of relatively prime numbers satisfying $a+b=c$ are called {\em $abc$ triples} if the product of their distinct prime divisors is strictly less than $c$. We catalog what is known about $abc$ triples, both numerical examples found through computation and infinite familes of examples established theoretically. In addition, we collect motivations and heuristics supporting the $abc$ conjecture, as well as some of its refinements and generalizations, and we describe the state-of-the-art progress towards establishing the conjecture.

We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.3))$. This is done by first deriving the Riemann--von Mangoldt explicit formula for the Riemann zeta-function with explicit bounds on the error term. We use this along with other recent explicit estimates regarding the zeroes of the Riemann zeta-function to obtain the result. Furthermore, we show that there is a prime between any two consecutive $m$th powers for $m \geq 5 \times 10^9$. Notably, many of the explicit estimates developed in this paper can also find utility elsewhere in the theory of numbers.

Let ${\mathcal P} \subset {\mathbf R}^n$ be a rational convex polytope with vertices at the origin and on each positive coordinate axes. On the basis of the study on counting lattice points in $t{\mathcal P}$ with positive integer $t$, which is deeply connected with reciprocity laws for generalized Dedekind sums, we study the number of lattice points in the shifted polytope of $t{\cal P}$ by a fixed rational point. Certain generalized multiple Dedekind sums appear naturally in the main result.

In this paper we examine Grosswald's conjecture on $g(p)$, the least primitive root modulo $p$. Assuming the Generalized Riemann Hypothesis (GRH), and building on previous work by Cohen, Oliveira e Silva and Trudgian, we resolve Grosswald's conjecture by showing that $g(p)< \sqrt{p} - 2$ for all $p>409$. Our method also shows that under GRH we have $\hat{g}(p)< \sqrt{p}-2$ for all $p>2791$, where $\hat{g}(p)$ is the least prime primitive root modulo $p$.

We show a relation between the Mordell-Tornheim multiple zeta-function and the confluent hypergeometric function, and using it, we give the functional equation for the Mordell-Tornheim multiple zeta-function. In the double case, the functional equation includes the known functional equation for the Euler-Zagier double zeta-function.

Let ${\tt d}$ be the density of a probability distribution that is compactly supported in the positive semi-axis. Under certain mild conditions we show that $$ \lim_{x\to\infty}x\sum_{n=1}^\infty \frac{{\tt d}^{*n}(x)}{n}=1,\qquad\text{where}\quad {\tt d}^{*n}:=\underbrace{{\tt d} *{\tt d}*\cdots*{\tt d}}_{n\text{~times}}. $$ We also show that if $c>0$ is a given constant for which the function $f(k):=\widehat{\tt d}(k)-1$ does not vanish on the line $\{k\in\mathbb{C}:\Im k=-c\}$, where $\widehat{\tt d}$ is the Fourier transform of ${\tt d}$, then one has the asymptotic expansion $$ \sum_{n=1}^\infty\frac{{\tt d}^{*n}(x)}{n}=\frac{1}{x}\bigg(1+\sum_k m(k) e^{-ikx}+O(e^{-c x})\bigg)\qquad (x\to +\infty), $$ where the sum is taken over those zeros $k$ of $f$ that lie in the strip $\{k\in\mathbb{C}:-c<\Im k<0\}$, $m(k)$ is the multiplicity of any such zero, and the implied constant depends only on $c$. For a given distribution of this type, we briefly describe the location of the zeros $k$ of $f$ in the lower half-plane $\{k\in\mathbb{C}:\Im k<0\}$. For an odd prime $p$, let $n_0(p)$ be the least natural number such that $(n|p)=-1$, where $(\cdot|p)$ is the Legendre symbol. As an application of our work on probability distributions, we generalize a well known result of Heath-Brown concerning the exhibited behavior of the Dirichlet $L$-function $L(s,(\cdot|p))$ under the assumption that the Burgess bound $n_0(p)\ll p^{1/(4\sqrt{e})+\varepsilon}$ cannot be improved.