Given an integer $q\ge 2$, a $q$-normal number is an irrational number $\eta$ such that any preassigned sequence of $k$ digits occurs in the $q$-ary expansion of $\eta$ at the expected frequency, namely$1/q^k$. In a series of recent papers, using the complexity of the multiplicative structure of integers along with a method we developed in 1995 regarding the distribution of subsets of primes in the prime factorization of integers, we initiated new methods allowing for the creation of large families of normal numbers. Here, we further expand on this initiative.

The partial Stirling numbers $T_n(k)$ used here are defined as $\sum_{i\text{ odd}}\binom{n}{i}i^k$. Their 2-exponents $\nu(T_n(k))$ are important in algebraic topology. We provide many specific results, applying to all values of $n$, stating that, for all $k$ in a certain congruence class mod $2^t$, $\nu(T_n(k))=\nu(k-k_0)+c_0$, where $k_0$ is a 2-adic integer and $c_0$ a positive integer. Our analysis involves several new general results for $\nu(\sum\binom{n}{2i+1}i^j)$, the proofs of which involve a new family of polynomials. Following Clarke [3], we interpret $T_n$ as a function on the 2-adic integers, and the 2-adic integers $k_0$ described above as the zeros of these functions.

Let $j_0 \mapsto (j_0, J_1(j_0), J_2(j_0), \ldots)$ be the map that takes the $j$-invariant of an ordinary elliptic curve in characteristic $p$ to the $j$-invariant of its canonical lifting over the ring of Witt vectors. We have that $J_i \in \mathbb{F}_p(X)$, and in this paper we describe how to derive these rational functions from the modular polynomial in an efficient way and give more precise description of the numerators and denominators of the reduced forms of these functions. In particular, upper bounds are given for the order of their poles.

In this paper, we give all the integer solutions of the equation $(x+1)^2+(x+2)^2+...+(2x)^2=y^n$.

The arithmetical nature of values at rational points of the hypergeometric series \[ _Q G_R(x) := \sum_{n=0}^\infty \frac{R(1) R(2) \cdots R(n)}{Q(1) Q(2) \cdots Q(n)} x^n \] is studied. $R$ and $Q$ are polynomials with integer coefficients. Using deep results on higher con\-gruences going essentially back to Frobenius, Dedekind, Nagell and Schinzel a measure of $\mathbb{Q}$-linear independence of such values is given. In contrast to former investigations no preseribed factorisations of the polynomials $Q$ and $R$ are necessary. Here, however, congruences to those primes $p$ are used for which $Q$ mod $p$ splits completely into a product of linear factors. To get the measure the fact is used that these primes have a Dirichlet--density. In six applications results, proven in some particular cases by different techniques by Carlson [2], Inkeri [15], Ivankov [17,18], Popken [26] and Bundschuh--Wallisser [1] are derived.

The $p$-adic diaphony as introduced by Hellekalek is a quantitative measure for the irregularity of distribution of a sequence in the unit cube. In this paper we show how this notion of diaphony can be interpreted as worst-case integration error in a certain reproducing kernel Hilbert space. Our main result is an upper bound on the $p$-adic diaphony of the Halton sequence.

In this paper, we give an explicit formula for computing Fourier coefficients of octahedral newforms of weight one associated to three-division points of elliptic curves.

Let $m_1$, $m_2$, and $m_3$ be distinct square-free integers (including $1$). First, we show that there exist infinitely many square-free integers $d$ with $\gcd(m_1m_2, d) = 1$ such that the class numbers of $\mathbb{Q}(\sqrt{m_1d})$ and $\mathbb{Q}(\sqrt{m_2d})$ are both divisible by $3$. This is a generalization of a result of T.~Komatsu [15]. Secondly, we show that there exist infinitely many positive fundamental discriminants $D$ with $\gcd(m_1m_2m_3, D) = 1$ such that the class numbers of real quadratic fields $\mathbb{Q}(\sqrt{m_1D})$, $\mathbb{Q}(\sqrt{m_2D})$, and $\mathbb{Q}(\sqrt{m_3D})$ are all indivisible by $3$ when $m_1$, $m_2$, and $m_3$ are positive. This is a generalization of a result of D.~Byeon [4]. We add an application of this result to the Iwasawa invariants related to Greenberg's conjecture.

It is a well known result that the number of points over a finite field on the Legendre family of elliptic curves can be written in terms of a hypergeometric function modulo $p$. In this paper, we extend this result, due to Igusa, to a family of monomial deformations of a diagonal hypersurface. We find explicit relationships between the number of points and generalized hypergeometric functions as well as their finite field analogues.

In this paper, we study the relation between the discrete and the continuous mean values of $\Delta_a^2(x)$, where $\Delta_a(x)$ $(-1<a<1)$ is the error term in the generalized divisor problem. We try to find the formula of the difference of these mean values in a sufficiently explicit form. As an application we give the asymptotic formula of the discrete mean square of $\Delta_a(n)$ in the range $-1<a<1, a\not=0$. We also study the integral containing the error term in the weighted two-dimensional divisor problem.

This paper is a continuation of [2], [9] and [10]. Its aim is to characterize non-removable ideals in the class ${\cal M}$ of all multiplicatively convex bornological algebras. Let $A$ be in ${\cal M}$. We show that an ideal $I\subset A$ is ${\cal M}$-non-removable if and only if it consists of joint m-bounding elements (D\'ef. 9). Such an ideal doesn't consist, in general, of joint bounding elements (D\'ef. 8). This shows that the concept of ${\cal M}$-non-removable ideals is of relative character (D\'ef. 4).