We prove a Khintchine type theorem for approximation of elements in the Cantor set, as a subset of the formal Laurent series over $\mathbb{F}_3$, by rational functions of a specific type. Furthermore we construct elements in the Cantor set with any prescribed irrationality exponent $\geq 2$.

We give an explicit upper bound for non-principal Dirichlet $L$-functions on the line $s=1+it$. This result can be applied to improve the error in the zero-counting formulae for these functions.

Let $K$ be an algebraic number field and $\nu_K$ be the ideal-counting function of $K$. Many authors have estimated the remainder term $\Delta_n(x,K)$ in the asymptotic formula of the average order of $\nu_K$. The purpose of this work is twofold: we first generalize Müller's method to the $n$-dimensional case and improve on Nowak's result. A key part in the proof is played by a~profound result on a triple exponential sum recently derived by Robert \& Sargos.

The Fibonacci-type numbers in the title look like $R_n=g_1\gamma_1^n+ g_2\gamma_2^n$ and $S_n=h_1\gamma_1^n+h_2\gamma_2^n$ for any $n\in\mathbb{Z}$, where the $g$'s, $h$'s, and $\gamma$'s are given algebraic numbers satisfying certain natural conditions. For fixed $k\in\mathbb{Z}_{>0}$, and for fixed non-zero periodic sequences $(a_h),(b_h),(c_h)$ of algebraic numbers, the algebraic independence of the series \[ \sum_{h=0}^\infty \frac{a_h}{\gamma_1^{kr^h}}\,, \quad {\sum_{h=0}^\infty}\,\strut' \frac{b_h}{(R_{kr^h+\ell})^m}\,, \quad {\sum_{h=0}^\infty}\,\strut' \frac{c_h}{(S_{kr^h+\ell})^m} \qquad \big((\ell,m,r)\in\mathbb{Z}\times \mathbb{Z}_{>0}\times\mathbb{Z}_{>1}\big) \] is studied. Here the main tool is Mahler's method which reduces the investigation of the algebraic independence of numbers (over $\mathbb{Q}$) to that of functions (over the rational function field) if they satisfy certain types of functional equations.

The number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, and Akatsuka. Berndt, Levinson, and Montgomery studied the general case, meanwhile Akatsuka gave sharper estimates for the first derivative of the Riemann zeta function under the truth of the Riemann hypothesis. In this paper, we generalize the results of Akatsuka to the $k$-th derivative (for positive integer $k$) of the Riemann zeta function.

Let $f$ be a holomorphic cusp form of weight $k$ with respect to the full modular group $SL_2(\mathbb{Z})$. We suppose that $f$ is a normalized Hecke eigenform. Let $L_f(s)$ be the $L$-function attached to the form $f$. Good gave the approximate functional equation and mean square formula of $L_f(s)$. In this paper, we shall generalize these formulas for the derivatives of $L_f(s)$.

In this paper, we show that any additive complex valued function over non-zero Gaussian integers which vanishes on the shifted Gaussian primes is necessarily identically zero.

We study a $q$-analogue of multiple zeta values that was proposed by Zudilin and is closely related to that of Schlesinger. We explore the double $q$-shuffle structure and provide an Euler decomposition formula. Furthermore we compare our results with the classical multiple zeta values and the $q$-models of Ohno-Okuda-Zudilin and Bradley.

Consider the diophantine equation $ax^{3} + by + c = xyz$, where $a, b$ and $c$ are positive integers such that $\gcd(a, c) = 1$ and $c$ is square-free. Let $(x, y, z)$ be a positive integral solution of the equation. In this paper, we shall give an upper bound for $x$,$y$ and $z$ in terms of the given inputs $a$, $b$ and $c$. Also, we apply our results to investigate the divisors of the elements of the sequence $\{an^{3} + c\}$ in residue classes.