In this paper we generalize some results, obtained by Shimura, Yoshida and the author, on critical values of $L$-functions of $l$-adic representations attached to Hilbert modular forms twisted by finite order characters, to the critical values of $L$-functions of arbitrary base change to totally real number fields of $l$-adic representations attached to Hilbert modular forms twisted by some finite-dimensional representations.

We study the analytic behavior of a power series with coefficients containing the von Mangoldt function. In particular, we extend an explicit formula of Hardy and Littlewood for related functions and derive further representation formulas in the unit disk that reveal logarithmic singularities on a dense subset of the unit circle. As an essential tool for proving the square integrability of occurring limit functions together with respective error estimates we contribute a new proof of a Ramanujan-like expansion of an arithmetic function consisting of the von Mangoldt function and the Euler function.

We consider the parametric family of sextic Thue equations $$x^6-2mx^5y-5(m+3)x^4y^2-20x^3y^3+5mx^2y^4+2(m+3)xy^5+y^6=\lambda$$ where $m\in\mathbb{Z}$ is an integer and $\lambda$ is a divisor of $27(m^2+3m+9)$. We show that the only solutions to the equations are the trivial ones with $xy(x+y)(x-y)(x+2y)(2x+y)=0$.

For a rational $q=u+\frac{\alpha}{d}$ with $u, \alpha, d\in \ACOBZ$ with $u\ge 0, 1\le \alpha<d$, $\gcd(\alpha, d)=1$, the \emph{generalized Hermite-Laguerre polynomials $G_q(x)$} are defined by \begin{align*} G_q(x)&=a_nx^n+a_{n-1}(\alpha +(n-1+u)d)x^{n-1}+\cdots\\ &\quad+a_1\left(\prod^{n-1}_{i=1}(\alpha +(i+u)d)\right)x+a_0 \left(\prod^{n-1}_{i=0}(\alpha +(i+u)d)\right) \end{align*} where $a_0, a_1, \cdots , a_n$ are arbitrary integers. We prove some irreducibility results of $G_q(x)$ when $q\in \{\frac{1}{3}, \frac{2}{3}\}$ and extend some of the earlier irreducibility results when $q$ of the form $u+\frac{1}{2}$. We also prove a new improved lower bound for greatest prime factor of product of consecutive terms of an arithmetic progression whose common difference is $2$ and $3$.

Let $p$ be a large prime, and let $C$ be a hyperelliptic curve over $\mathbb{F}_p$. We study the distribution of the $x$-coordinates in short intervals when the $y$-coordinates lie in a prescribed interval, and the distribution of the distance between consecutive $x$-coordinates with the same property. Next, let $g(P,P_0)$ be a rational function of two points on $C$. We study the distribution of the above distances with an extra condition that $g(P_i,P_{i+1})$ lies in a prescribed interval, for any consecutive points $P_i,P_{i+1}$.

Levinson and Montgomery in 1974 proved many interesting formulae on the zeros of derivatives of the Riemann zeta function $\zeta(s)$. When Conrey proved that at least 2/5 of the zeros of the Riemann zeta function are on the critical line, he proved the asymptotic formula for the mean square of $\zeta(s)$ multiplied by a mollifier of length $ T^{4/7}$ near the $1/2$-line. As a consequence of their papers, we study some aspects of zeros of the derivatives of the Riemann zeta function with no assumption.

Let $E$ be a CM elliptic curve defined over an algebraic number field $F$ with CM by an imaginary quadratic field $K$. We determine the group of $K_{ab}F$-rational torsion points of $E$. In some cases we also determine the group of $F$ or $KF$-rational torsion points of $E$.

For each prime number $\ell$ less than $10^4$, we construct an infinite family of abelian number fields for which Iwasawa $\lambda_{\ell}$-invariants vanish.

Let $\alpha$ be an algebraic integer and assume that it is {\it expanding}, i.e., its all conjugates lie outside the unit circle. We show several results of the form $\mathbb{Z}[\alpha]=\mathcal{B}[\alpha]$ with a certain finite set $\mathcal{B}\subset\mathbb{Z}$. This property is called {\it height reducing property}, which attracted special interest in the self-affine tilings. Especially we show that if $\alpha$ is quadratic or cubic trinomial, then one can choose $\mathcal{B}= \left\{0,\,\pm 1,\,\ldots,\,\pm \left(|N(\alpha)|-1\right)\right\}$, where $N(\alpha)$ stands for the absolute norm of $\\alpha$ over $\mathbb{Q}$.

In previous papers the authors established a method how to decide on the algebraic independence of a set $\{ y_1,\dots ,y_n \}$ when these numbers are connected with a set $\{ x_1,\dots ,x_n \}$ of algebraic independent parameters by a system $f_i(x_1,\dots ,x_n,y_1,\dots ,y_n) =0$ $(i=1,2,\dots ,n)$ of rational functions. Constructing algebraic independent parameters by Nesterenko's theorem, the authors successfully applied their method to reciprocal sums of Fibonacci numbers and determined all the algebraic relations between three $q$-series belonging to one of the sixteen families of $q$-series introduced by Ramanujan. In this paper we first give a short proof of Nesterenko's theorem on the algebraic independence of $\pi$, $e^{\pi\sqrt{d}}$ and a product of Gamma-values $\Gamma (m/n)$ at rational points $m/n$. Then we apply the method mentioned above to various sets of numbers. Our algebraic independence results include among others the coefficients of the series expansion of the Heuman-Lambda function, the values $P(q^r), Q(q^r)$, and $R(q^r)$ of the Ramanujan functions $P,Q$, and $R$, for $q\in \overline{\ACADQ}$ with $0<|q|<1$ and $r=1,2,3,5,7,10$, and the values given by reciprocal sums of polynomials.