For integers $a_0,a_1,\ldots,a_n$ with $|a_0a_n|=1$ and either $\alpha =u$ with $1\leq u \leq 50$ or $\alpha=u+ \frac{1}{2}$ with $1 \leq u \leq 45$, we prove that $\psi_n^{(\alpha)}(x;a_0,a_1,\cdots,a_n)$ is irreducible except for an explicit finite set of pairs $(u,n)$. Furthermore, all exceptions other than $n=2^{12},\alpha=89/2$ are necessary. The above result with $0\leq\alpha \leq 10$ is due to Filaseta, Finch and Leidy and with $\alpha \in \{-1/2,1/2\}$ due to Schur.

Let $f\in\mathbb{Z}[x_1,\ldots,x_N]$ be a nonzero polynomial. We show that there exists a sequence of real numbers defined in terms of the coefficients of $f$, converging to the Mahler's measure of $f$.

For a fixed $\theta\neq 0$, we define the twisted divisor function \[ \tau(n, \theta):=\sum_{d\mid n}d^{i\theta}\ . \] In this article, we consider the error term $\Delta(x)$ in the following asymptotic formula \[ \sum_{n\leq x}|\tau(n, \theta)|^2=\omega_1(\theta)x\log x + \omega_2(\theta)x\cos(\theta\log x) +\omega_3(\theta)x + \Delta(x),\] where $\omega_i(\theta)$ for $i=1, 2, 3$ are constants depending only on $\theta$. We obtain \[\Delta(T)=\Omega\left(T^{\alpha(T)}\right) \text{ where } \alpha(T) =\frac{3}{8}-\frac{c}{(\log T)^{1/8}} \text{ and } c>0,\] along with an $\Omega$-bound for the Lebesgue measure of the set of points where the above estimate holds.

Using the theory of elliptic curves, we study the nontrivial rational (parametric) solutions of the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ for some simple Laurent polynomials $f(x)$.

In this article, we discuss several aspects of convergence theorems for nets and sequences of Riemann and Riemann-type integrable functions defined on a closed bounded interval in $\mathbb{R}$ with values in a Banach space. We introduce the notions of Riemann $\Delta$-Cauchy nets of functions with its analogous variants and derive some correlations between such kind of nets of functions and equi-Riemann integrability. Moreover, we establish equi-integrability of the pointwise closure of different types of equi-integrable collections of functions. Finally, several related results, e.g., relative compactness of equi-integrable collections of functions with respect to different topologies are studied.

We generalize Chen's theorem [2] $$\displaystyle\sum_{r=1}^{m}\sum_{|\boldsymbol{\alpha}|=m}{m\choose\boldsymbol{\alpha}}\zeta(\boldsymbol{\alpha}n)=\zeta^{m}(n)$$ for complex arguments, presenting a very elementary proof. Subsequently, using a similar technique, we obtain a general formula that allows us to prove relations between nonlinear Euler sums. We also present relations between series of powers of the Hurwitz zeta function and the multiple zeta function, like\vspace{-1pt} $$ \displaystyle\sum_{n=1}^{\infty}\zeta_{H}^{m}(s,n)=\sum_{r=1}^{m}\sum_{|\boldsymbol{\alpha}|=m}{m\choose\boldsymbol{\alpha}}\zeta(\alpha_{1}s-1,\alpha_{2}s,\dots,\alpha_{r}s).$$

We prove that $$ \int_1^X\Delta(x)\Delta_3(x)dx \ll X^{13/9}\log^{10/3}X, \qquad \int_1^X\Delta(x)\Delta_4(x)\d x \ll_\varepsilon X^{25/16+\varepsilon}, $$ where $\Delta_k(x)$ is the error term in the asymptotic formula for the summatory function of $d_k(n)$, generated by $\zeta^k(s)$ ($\Delta_2(x) \equiv \Delta(x)$). These bounds are sharper than the ones which follow by the Cauchy-Schwarz inequality and mean square results for $\Delta_k(x)$. We also obtain the analogues of the above bounds when $\Delta(x)$ is replaced by $E(x)$, the error term in the mean square formula for $|\zeta(1/2+it)|$.

In this paper we propose a new family of nonuniform random recursive trees. The probability of joining new vertex to the vertex $i$ depends only on $i$ (and the size of the tree). We deal with the degrees of vertices of such trees.