For a compositum $k$ of quadratic number fields new explicit units are constructed by taking power-of-two roots of circular units. These units are used to obtain a result concerning the divisibility of the class number of $k$ by a power of $2$.

Let $p$ be a fixed prime number. Let $S_k(N)$ be the space of cusp forms of weight $k$ and level $N$. We prove a weighted equidistribution theorem for the eigenvalues of the $p$-th Hecke operator $T_p$ acting on $S_k(N)$. This is a variant of a celebrated theorem of Serre.

Let $f \in \mathbb{Z}[X,Y,Z]$ be a non-constant, absolutely irreducible, homogeneous polynomial with integer coefficients, such that the projective curve given by $f=0$ has a~function field isomorphic to the rational function field $\mathbb{Q} (T)$. We show that all integral solutions of the Diophantine equation $f=0$ (up to those corresponding to some singular points) can be parametrized by a single triple of integer-valued polynomials. In general, it is not possible to parametrize this set of solutions by a~single triple of polynomials with integer coefficients.

We discuss hypergeometric constructions of rational approximations to values of the logarithm function.

Rational $p$-adic zeros of the Leopoldt-Kubota $p$-adic $L$-functions give rise to certain sequences of generalized Bernoulli numbers tending $p$-adically to zero, and conversely. This relationship takes different forms depending on whether the corresponding Iwasawa $\lambda$-invariant is one or greater than one. To understand the relationship better it is useful to consider approximate zeros of those functions.

A recent development of the Davenport-Heilbronn method for diophantine inequalities is reexamined, and then applied to a class of problems in diophantine approximation. Among other things, an asymptotic formula is obtained for the number of solutions of the simultaneous inequalities $|n_j - \lambda_j n_0| <\varepsilon$ with square-free $n_j \in [1,N]$, whenever the positive real numbers $\lambda_1, \ldots, \lambda_r$ and $1$ are linearly independent over the rationals.

Let $\theta$ be a Salem number. It is well-known that the sequence $(\theta^n)$ modulo 1 is dense but not equidistributed. In this article we discuss equidistributed subsequences. Our first approach is computational and consists in estimating the supremum of $\lim_{n\to\infty} n/s(n)$ over all equidistributed subsequences $(\theta^{s(n)})$. As a result, we obtain an explicit upper bound on the density of any equidistributed subsequence. Our second approach is probabilistic. Defining a measure on the family of increasing integer sequences, we show that relatively to that measure, almost no subsequence is equiditributed.

We determine how big an Euler product can be at $s_2$, when its size at $s_1$ is known, and apply this via Halász's method to bound the mean value of a multiplicative function in terms of the size of the generating Dirichlet series.

In this paper we continue our investigations concerning arithmetic graphs associated with integral domains and their applications to diophantine problems. We establish some general quantitative theorems for these graphs considered over finitely generated integral domains and prove some effective analogues over number fields and function fields. Further, we apply our results to resultant equations and discriminant equations. In a separate paper, further applications will be given to decomposable form equations, algebraic numbers and irreducible polynomials.

Let us denote by $\tau(n)$ and $\sigma(n)$ the number and the sum of the divisors of $n$ and by $\varphi$ Euler's function. We give effective upper bounds for $\frac{n}{\varphi(n)}$ in terms of $\varphi(n)$, and for $\frac{\sigma(n)}{n}$ in terms of $\tau(n)$.

In this paper, we show that the sequences of arithmetic, geometric and harmonic means of the first $n$ values of the Euler function are all three dense modulo $1$. This answers in the affirmative a question of A. Schinzel.

A fortuitous intersection of work on periodic continued fraction expansions in hyperelliptic function fields and the study of parametrized families of quadratic number fields with high class number leads us to discover sequences of hyperelliptic curves whose Jacobians contain torsion divisors of order $g^2$. These sequences generalize those earlier constructed by Flynn and by Leprévost.