Let $\omega = \exp (2\pi i/3)$. In this paper, we study moments of central values of sextic Hecke $L$-functions of $\mathbb{Q}(\omega)$ and one level density result for the low-lying zeros of sextic Hecke $L$-functions of $\mathbb{Q}(\omega)$. As a corollary, we deduce that, assuming GRH, at least $2/45$ of the members of the sextic family do not vanish at $s=1/2$.

We discuss the triviality of some classical Iwasawa modules in connection with the notion of $\ell$-rationality for totally $\ell$-adic number fields.

In this paper we compute an explicit upper bound of polynomial typefor the size of the smallest integral solution of the Pell equation over a number field of degree $\geq 2$.

First we reprove two results in additive number theorydue to Dombi and Chen & Wang, respectively, on the number ofrepresentations of $n$ as the sum of two odious or evil numbers, using techniques from automata theory and logic. We also use this technique to prove a new result aboutthe numbers represented by five summands. Furthermore, we prove some new results on the tenfold sums of the evil and odious numbers, as well as $k$-fold sums of similar sequences of integers, by using techniques of analytic number theory involving trigonometric sums associated with the $\pm 1$ characteristic sequences of these integers.

In 1977 Pohst conjectured a certain inequality for $n$ variables and gave a computer-assisted proof for $n\leq 10$. We give a proof for all $n$ using a combinatorial argument. This inequality yields a better bound for the regulator in terms of the discriminant for totally real number fields.

Monogenity of pure fields is recently intensively investigated. There are several papers on conditions for the monogenity and non-monogenity of pure fields of various degrees.However, efficient algorithms for determining generators of power integral basesexists only for pure cubic and pure quartic fields.In our paper we present an efficient method to determine all generators of power integral bases of pure sextic fields.

We evaluate the twisted first moment of central values of the product of a quadratic Dirichlet $L$-function and a quadratic twist of a modular $L$-function.

Let $q$ be a prime. We give an elementary proof of the fact that for any $k\in\mathbb{N}$, the proportion of $k$-element subsets of $\mathbb{Z}$ that contain a $q^{\text{th}}$ power modulo almost every prime, is zero. This result holds regardless of whether the proportion is measured additively or multiplicatively. More specifically, the number of $k$-element subsets of $[-N, N]\cap\mathbb{Z}$ that contain a $q^{\text{th}}$ power modulo almost every prime is no larger than $a_{q,k} N^{k-(1-\frac{1}{q})}$, for some positive constant $a_{q,k}$. Furthermore, the number of $k$-element subsets of $\{\pm p_{1}^{e_{1}} p_{2}^{e_{2}} \cdots p_N^{e_N} : 0 \leq e_{1}, e_{2}, \ldots, e_N\leq N\}$ that contain a $q^{\text{th}}$ power modulo almost every prime is no larger than $m_{q,k} \frac{N^{Nk}}{q^N}$ for some positive constant $m_{q,k}$.