In this paper we prove that the shifted Hermite polynomial $H_{n}(x)+b$ has at least three simple zeros for each complex number $b$, provided that $n\geq 7$.

We prove new identities for the constant in the Mertens product over primes in the arithmetic progressions $a \bmod q$.

Let $a_1>a_2>1$ and $b_1>b_2>1$ be two distinct pairs of multiplicatively independent integers. If $b_1>a_1$ and $a_2>b_2$ or $b_1<a_1$ and $a_2<b_2$ then we prove that for every $\xi_1,\xi_2,$ with at least one $\xi_i$ irrational, there exists $q\in\mathbb{N}$ such that for any sequence of real numbers $r_m$ the set of sums \[ \{a_1^ma_2^{n}q\xi_1+b_1^mb_2^{n}q\xi_2+r_m:m,n\in\mathbb{N}\}, \] is dense modulo 1. The sets with algebraic numbers $a_i,b_i$ are also considered.

We present an algorithm producing all rational functions $f$ with prescribed $n+1$ Taylor coefficients at the origin and such that $||f||_\infty \le 1$ and $\deg f \le k$ for every fixed $k\ge n$. The case where $k<n$ is also discussed.

In this note, we define a $q$-analogue of the Mahler measures by using the Jackson integral which we call the Jackson $q$-Mahler measures. Especially we study their classical limit for polynomials of one variable.

In the present paper we prove that all three distinct triangular numbers in geometric progression and the positive integer solutions $(x,y,z)$ of the equation $(x^2-1)(y^2-1)= (z^2-1)^2$, $1<x<z<y$, $2\not|xyz$ are one-to-one under the assumption that a conjecture on a system of diophantine equations holds.

Let $k$ be a natural number, $k \ge 3$. Let $V_k(x)$ be the number of solutions $(u, v)$ of \[|u|^k + |v|^k \le x,\qquad (u, v) = 1\] and let \[E_k(x) = V_k(x) - \frac{3\Gamma^2(1/k)}{\pi^2\Gamma(2/k)} x^{2/k}.\] Under the Riemann hypothesis, it is known that \[E_k(x) = O(x^{\t_k+\e})\] for $k = 3, 4$, where $\t_3 = \frac{9581}{36864}$ and $\t_4 = \frac{107}{512}$. The result for $k=3$ is shown to require only a~zero-free strip $Re s > 0.5802\ldots$ for $\z(s)$. For $k=4$, the exponent is improved by about 0.0016. Here the required zero-free strip is $Re s > 0.7058\ldots$. Along the way, new results are obtained for the mean value on a vertical segment $[\s, \s + iT]$ of the Hlawka zeta function defined by \[Z_k(s) = \sum_{n=1}^\infty \frac{r_k(n)}{n^s} \qquad (Re s >1),\] where $r_k(n) = \#\{(x, y) : |x|^k + |y|^k = n\}$. These mean value results improve those in the literature for all $k\ge 3$. The mean value of $E_k(x)$ is also considered, and here an asymptotic formula is provided under the assumption of a zero-free strip of width $> 1/k$. Previous writers required $k \ge 5$ and the Riemann hypothesis.