We obtain in this paper the solutions of the difference equations $$ x_{n+1}=\dfrac{ax_{n}x_{n-2}}{x_{n-1}(-b+cx_{n}x_{n-2})},\qquad n=0,1,..., $$ where $a$, $b$, $c$ are positive real numbers and the initial conditions are arbitrary positive real numbers.

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.

Using techniques due to Coster, we prove a supercongruence for a generalization of the Domb numbers. This extends a recent result of Chan, Cooper and Sica and confirms a~conjectural supercongruence for numbers which are coefficients in one of Zagier's seven ``sporadic'' solutions to second order Apéry-like differential equations.

We prove that the upper bound for the van der Corput property of the set of shifted primes is $O((\log n)^{-1+o(1)})$, giving an answer to a problem considered by Ruzsa and Montgomery for the set of shifted primes $p-1$. We construct normed non-negative valued cosine polynomials with the spectrum in the set $p-1$, $p\leq n$, and a small free coefficient $a_{0}=O((\log n)^{-1+o(1)})$. This implies the same bound for the intersective property of the set $p-1$, and also bounds for several properties related to uniform distribution of related sets.

We obtained sufficient conditions for existence of fixed points of involutions in CAT$(0)$ spaces. Convergence results of Mann and Ishikawa iterates of weakly contractive mappings are also proved.

We consider Mertens' function in arithmetic progression, \[ M(x,q,a) := \sum{n\le x, n\equiv a mod q} \mu(n). \] Assuming the generalized Riemann hypothesis (GRH), we show that the bound \[ M(x,q,a)\ll_{\varepsilon} \sqrt{x}\exp{((\log x)^{3/5}(\log\log x)^{16/5 +\varepsilon})} \] holds uniform for all $q\le \exp(\frac{\log 2}{2}\lfloor (\log x)^{3/5}(\log\log x)^{11/5}\rfloor)$, $\gcd(a,q)=1$ and all $\varepsilon>0$. The implicit constant is depending only on $\varepsilon$. For the proof, a former method of K. Soundararajan is extended to $L$-series.

We observe that five polynomial families have all of their roots on the unit circle. We prove the statements explicitly for four of the polynomial families. The polynomials have coefficients which involve Bernoulli numbers, Euler numbers, and the odd values of the Riemann zeta function. These polynomials are closely related to the Ramanujan polynomials, which were recently introduced by Murty, Smyth and Wang [MSW]. Our proofs rely upon theorems of Schinzel [S], and Lakatos and Losonczi [LL] and some generalizations.

In the paper [2] we have shown that in the case of a separable Banach algebra $A$ the necessary and sufficient condition in order that a closed ideal $I\subset A$ has a dense subideal is that $I$ is not finitely (algebraically) generated. We conjectured that this result is true in the general case. In this paper we give an example showing that this conjecture fails to be true.

The paper builds on earlier published work by the author in which a measure for the slowness of clustering of a bounded real sequence, called \emph{separation}, was introduced. Here a conjecture of the earlier paper is proved: that a particular sequence of rational numbers -- the $\mathrm{f}$ sequence -- defined in that paper is of maximal separation.

Let $N$ denote a sufficiently large odd integer. In this paper it is proved that $N$ can be represented as the sum of three primes, one of which is $\leq N^{\frac{11}{400}+\varepsilon}$ for any $\varepsilon>0$. This result constitutes an improvement upon that of K.C. Wong, who obtained the exponent $\frac{7}{216}$.

Given an integer $t\ge 1$, a rational number $g$ and a prime $p\equiv 1 (mod t)$ we say that $g$ is a near-primitive root of index $t$ if $\nu_p(g)=0$, and $g$ is of order $(p-1)/t$ modulo $p$. In the case $g$ is not minus a square we compute the density, under the Generalized Riemann Hypothesis (GRH), of such primes explicitly in the form $\rho(g)A$, with $\rho(g)$ a rational number and $A$ the Artin constant. We follow in this the approach of Wagstaff, who had dealt earlier with the case where $g$ is not minus a square. The outcome is in complete agreement with the recent determination of the density using a very different, much more algebraic, approach due to Hendrik Lenstra, the author and Peter Stevenhagen.

Let $\mathbb{F}_{q^e}$ be a finite field, and let $\mathbb{F}_{q^d}$ be a subfield of $\mathbb{F}_{q^e}$. The \emph{value set} of a polynomial $f$ lying within $\mathbb{F}_{q^d}$ is defined as the set of images $\{f(c) \in \\mathbb{F}_{q^d}\colon c \in \mathbb{F}_{q^e}\}$. This work is concerned with the cardinality of value sets of polynomials lying within subfields.