Let $\mathbf{k}$ be an arbitrary field. For any fixed badly approximable power series $\Theta$ in $\mathbf{k}((X^{-1}))$, we give an explicit construction of continuum many badly approximable power series $\Phi$ for which the pair $(\Theta, \Phi)$ satisfies the Littlewood conjecture. We further discuss the Littlewood conjecture for pairs of algebraic power series.

Infinite series representations are now obtained for certain weighted Rogers-Ramanujan partitions which we recently showed are related to partitions into $\mathrm{parts} \not\equiv 0$, $\pm i$ ($\mathrm{mod}\ 6$), for $i = 1, 2, 3$. We also show that our series can be transformed to the series previously obtained by Bressoud which connect the partitions into $\mathrm{parts} \not\equiv 0 \pm i$ ($\mathrm{mod}\ 6$) with partitions satisfying certain bounds on their successive ranks. Finally, we obtain finite versions of our series representations, namely, polynomial identities which tend to the infinite series identities when certain parameters tend to infinity.

A family of measures on the set of permutations of the first $n$ integers, known as Ewens sampling formula, arises in population genetics. In a series of papers, the first two authors have developed necessary and sufficient conditions for the weak convergence of a partial sum process based on these measures to a process with independent increments. Under very general conditions, it has been shown that a partial sum process converges weakly in a function space if and only if a related process defined through sums of independent random variables converges. In this paper, a functional limit theory is developed where the limiting processes need not be processes with independent increments. Thus, under Ewens sampling formula, the limiting process of the partial sums of dependent variables differs from that of the associated process defined through the partial sums of independent random variables.

Harmonic analysis and the elementary geometry of Hilbert spaces enable the representation of rationals by quotients of doubly-shifted primes. These representations offer an approach to lower bounds on the gaps between successive primes.

We prove the law of the iterated logarithm for gap series under weak regularity condition and apply to the lacunary trigonometric product.

Backlund showed that the Lindelöf hypothesis (LH) for the Riemann zeta-function is equivalent to some regularity of the distribution of zeros to the right from the critical line. We generalize the Backlund equivalent, by showing that LH is equivalent to the same type regularity of the distribution of any fixed complex value (not only zero). This generalized Backlund equivalent also can be applied for the Lerch zeta-function and, in our opinion, supports the idea that the Lindelöf hypothesis also is reasonable for zeta functions without the Euler product (usually having zeros off the critical line). Further we show that this generalized Backlund equivalent for LH can be formulated for zeta functions of the Selberg class and for the Selberg zeta-function, for which the Riemann hypothesis is true.

We present measure-theoretical dynamical systems behind certain types of digital sequences. This kind of result is a prerequisite if one sets out to study harmonic properties for this type of sequences with the method of skew-product transformations.

For primes $p$ let $$ A_p := \{n : n \in \mathbb{N},\ p|n\} $$ be the set of all natural numbers divisible by $p$. In his book "Probabilistic methods in the Theory of Numbers" (1964) J. Kubilius applies finite probabilistic models to approximate independence of the events $A_p$. His models are constructed to mimic the behaviour of (truncated) additive functions by suitably defined independent random variables.

Embedding $\mathbb{N}$, endowed with the discrete topology, in the compact space $\beta \mathbb{N}$, the Stone-Čech compactification of $\mathbb{N}$, and taking $\overline{A_p} := \mathrm{clos}_{\beta \mathbb{N}} A_p$ leads to independent events $\overline{A_p}$. This observation is a motivation for a general integration theory on $\mathbb{N}$ which can be used in various topics of Probabilistic Number Theory. In this paper we present a short compendium of Probabilistic Number Theory concerning the distribution of arithmetical functions. The new model is applied to the result of Erdös and Wintner about the limit distribution of additive functions and to the famous result of Szemeredi in combinatorical number theory. Further applications are given with respect to spaces of limit periodic and almost periodic functions and recent results on q-multiplicative functions.

For each $(\alpha, \beta)$ which is a periodic point related to modified Jacobi-Perron algorithm and $\mathbb{Q}(\alpha)$ has a complex embedding, we claim the following facts: the limit set of $\{(\sqrt{q_n} (q_n \alpha - p_n), \sqrt{q_n} (q_n \beta - r_n) | n = 1, 2, \dots\}$ is a finite union of similar ellipses, where $(p_n, q_n, r_n)$ is the $n$-th convergent $(p_n/q_n, r_n/q_n)$ of $(\alpha, \beta)$ by the modified Jacobi-Perron algorithm but for some $(\alpha, \beta)$ the ellipse given above is not the nearest ellipse in the limit set of $\{(\sqrt{q} (q\alpha - p), \sqrt{q} (q\beta - r) | q \in \mathbb{Z}, q \gt 0\}$ which is a union of similar ellipses.

A limit theorem in the sense of weak convergence of probability measures in the space of meromorphic functions for the Mellin transform of the square of the Riemann zeta-function is obtained.

We evaluate the speed of convergence in the Bohr-Jessen type of limit theorem on the value-distribution of Dedekind zeta-functions of number fields. When $K$ is a Galois number field, the Euler product of the corresponding Dedekind zeta-function $\zeta_K(s)$ is convex, hence the evaluation can be done similarly to the case of the Riemann zeta-function. However, when $K$ is non-Galois, some new ideas (based on the Artin-Chebotarev density theorem etc) are necessary, because the corresponding $\zeta_K(s)$ is not always convex.

A classical approach to study properties of $Q$-multiplicative functions $f(n)$ is to associate to the mean $\frac{1}{x} \sum_{0 \le n \le x} f(n)$ the product $\prod_{0 \le j \le k} \frac{1}{q_j} \sum_{0 \le a \le q_j-1} f(aQ_j)$. We discuss its validity in the case of non-negative $Q$-multiplicative functions $f(n)$ with a positive upper meanvalue, defined via a Cantor numeration system.

This is a survey of papers([3],[7],[8],[10],[11]). Uniformly distributed sequences generated by dynamical systems are considered. Relations between discrepancies and the spectra of the associated Perron-Frobenius operator are discussed. Examples of two and three dimensional low discrepancy sequences are also given.

The Rauzy induction is a dynamical system acting on the space of interval exchange transformations which is introduced by Rauzy in [20] and used by Veech to give an affirmative answer to the Keane Conjecture in [24] and [25]. The results in [25] enable us to construct induced transformations and jump transformations to the various sets. In this article the dynamical systems obtained by composing some of those transformations are called renormalized Rauzy inductions. Note that the two fold iteration of continued fraction transformation can be regarded as a classical example of renormalized Rauzy inductions via appropriate conjugacy. Our present goal is to establish the same kinds of central limit theorems as obtained in [17] for a class of renormalized Rauzy inductions.

We establish the universality theorem on the strip $\{ s \in \mathbb{C} | 1/2 \lt \mathrm{Re}\ s \lt 1\}$ for automorphic $L$-functions attached to Maass forms for $SL(2, \mathbb{Z})$, without the assumption of the Ramanujan conjecture. From this theorem, some results concerning the value distribution of the derivatives of those $L$-functions are obtained.

Let $\alpha$ be an irrational number with diophantine approximation properties and let $\ell (z)$ be a logarithmic-like function. We study the diaphony $F_N$ of the sequence $(\alpha n + \ell (n))_{n \ge 1}$. As an example of our result, we show that if $\alpha$ has the bounded partial quotients of the continued fraction expansion and $\beta$ is non-zero real, then the sequence $(x_n)_{n \ge 1} = (\alpha [(n+1)/2] + (-1)^{n+1} \beta \log ([(n+1)/2]))_{n \ge 1}$ satisfies $N^{-\frac{3}{4}-\varepsilon} \ll F_N (x_n) \ll N^{-\frac{2}{3}}$ for any $0 \lt \varepsilon \lt 1/4$. In our proof, Atkinson's saddle-point lemma is very useful.

We give a survey about the topics mentioned in the title, with a more detailed description of the recent joint results of Goldston, Yıldırım and the author about small gaps between consecutive primes.

In this survey lecture it is intended to sketch some parts [chosen according to the author's interests] of the [early] history of Probabilistic Number Theory, beginning with Paul Turáns proof (1934) of the Hardy—Ramanujan result on the “normal order” of the additive function $\omega (n)$, the Erdős–Wintner Theorem, and the Erdős–Kac Theorem. Next, mean–value theorems for arithmetical functions, and the Kubilius model and its application to limit laws for additive functions will be described in short.

Subsuming applications of the theory of almost–periodic functions under the concept of "Probabilistic Number Theory", the problem of "uniformly–almost–even functions with prescribed values" will be sketched, and the Knopfmacher – Schwarz – Spilker theory of integration of arithmetical functions will be sketched. Next, K.–H. Indlekofers elegant theory of integration of functions $\mathbb{N} \to \mathbb{C}$ of will be described.

Finally, it is tried to scratch the surface of the topic "universality", where important contributions came from the university of Vilnius.

Assuming the Riemann hypothesis and two conjectures from random matrix theory, we prove that $$ \lambda = \limsup_{n \to \infty} (\gamma_{n+1} - \gamma_n) \frac{1}{2\pi} \log \frac{\gamma_n}{2\pi} = \infty. $$

We prove that for the uniquely ergodic $\mathbb{R}^d$-action associated with a primitive substitution tiling of finite local complexity, every measurable eigenfunction coincides with a continuous function almost everywhere. Thus, topological weak-mixing is equivalent to measure-theoretic weak-mixing for such actions. If the expansion map for the substitution is a pure dilation by $\theta \gt 1$ and the substitution has a fixed point, then failure of weak-mixing is equivalent to $\theta$ being a Pisot number.

By means of the adelic compactification $\widehat{R}$ of the polynomial ring $R := \mathbb{F}_q [x]$, $q$ being a prime, we give a probabilistic proof to a density theorem: $$ \frac{\# \{(m, n) \in \{0, 1, \dots, N-1\}^2\ ;\ \varphi_m \text{ and }\varphi_n \text{ are coprime}\}}{N^2} \to \frac{q-1}{q}, $$ as $N \to \infty$, for a suitable enumeration $\{\varphi_n\}_{n=0}^{\infty}$ of $R$. Then establishing a maximal ergodic inequality for the family of shifts $\{\widehat{R} \ni f \mapsto f + \varphi_n \in \widehat{R}\}_{n=0}^{\infty}$, we prove a strong law of large numbers as an extension of the density theorem.

In this paper, we will give a certain formula for the Riemann zeta function that expresses the Riemann zeta function by an infinte series consisting of $K$-Bessel functions. Such an infinite series expression is regarded as an analogue of the Chowla-Selberg formula. Roughly speaking, the Chowla-Selberg formula is the formula that expresses the Epstein zeta-function by an infinite series consisting of $K$-Bessel functions. In addition, we also give certain analogues of the Chawla-Selberg formula for Dirichlet $L$-functions and $L$-functions attached to holomorphic cusp forms. Moreover, we introduce a two variable function which is analogous to the real analytic Eisenstein series and give a certain limit formula for this one. Such a limit formula is regarded as an analogue of Kronecker's limit formula.

We prove the existence of a mean value of arithmetical functions of two variables : $\lim_{x, y \to \infty} (xy)^{-1} \sum_{m \le x, n \le y} f(m, n)$ under some conditions, and, when $f$ is a multiplicative function of two variables, express the mean value as an infinite product over all primes. Five examples are given which are not obtained by trivial generalizations of results on arithmetical functions of one variable.

A complex valued function $f : G \to \mathbb{C}$ on a topological group is called a Hartman function if there is a compact group $C$, a continuous homomorphism $\iota : G \to C$ with image $\iota (G)$ dense in $C$ and a function $F : C \to \mathbb{C}$ which is integrable in the Riemann sense and satisfies $F \circ \iota = f$. $H \subseteq G$ is called a Hartman set if the characteristic function $\mathbb{1}_H$ is a Hartman function. In the case $G = \mathbb{Z}$ such a function is a two sided infinite binary sequence, called a Hartman sequence.

The investigation of such objects is motivated by the interest in finitely additive invariant probability measures on groups as well as by questions from symbolic dynamics in the context of ergodic group rotations. Connections to number theory ($n\alpha$- and Beatty sequences), combinatorics (complexity of words), geometry (projection bodies of convex polytopes), dynamics (ergodic group rotations, Sturmian sequences), topology (compactifications, group topologies, in particular precompact ones) and harmonic analysis (almost periodicity and weak almost periodicity) have been studied recently. This note is an extended version of a survey talk on these topics.

Integral representations of $q$-analogues of the Barnes multiple zeta functions are studied. The integral representation provides a meromorphic continuation of the $q$-analogue to the whole plane and describes its poles and special values at non-positive integers. Moreover, for any weight, employing the integral representation, we show that the $q$-analogue converges to the Barnes multiple zeta function when $q \uparrow 1$ for all complex numbers.