The spectral large sieve inequality due to H. Iwaniec, that is an estimate for the mean square over a spectral interval of a linear form in the Fourier coefficients of Maass wave forms is reported by use of a formula of Y. Motohashi and the “hybrid” large sieve inequality. Then the result is generalized to the case, where the coefficients of the linear form may depend on the respective eigenvalue of the hyperbolic Laplacian, and also on a well-spaced set of points.

Basing on conjectures given by H. Cohen, H.W. Lenstra, Jr. and J. Martinet [2], [3], [4], [5] concerning the heuristics on class groups of number fields we deduce some quantitative conjectures on the statistical behaviour of orders of the tame kernel $K_{2}\mathcal{O}_{F}$ of the ring $\mathcal{O}_{F}$ of integers of quadratic number fields $F$ of discriminants $D, \vert D \vert \leq x$.

We investigate the number of $D$'s such that for $F = \mathbb{Q}(\sqrt{D})$ the order of $K_{2}\mathcal{O}_{F}$ is divisible by $3$.

In this paper we study the distribution in the arithmetic progressions (modulo a product of two primes) of reducible quadratic polynomials $(an+b)(cn+d)$ in short intervals, i.e. when $n \in [x, x + H], H = o(x)$; here $H = x^{\vartheta}$, with $\vartheta \in ]3/4, 1[$. Using Large Sieve techniques we get results beyond the classical level $\vartheta$, reaching $3\vartheta - 3/2$; these also improve the results of Salerno and Vitolo [6] in “large” intervals $(\vartheta = 1)$ obtaining level $3/2$ instead of $4/3$.

Theorem on four types of pseudoprimes with respect to Lucas sequences are proved.

If $n$ is an Euler-Lucas pseudoprime with parameters $P$ and $Q$ and $n$ is an Euler pseudoprime to base $Q, (n, P) = 1$, then $n$ is Lucas pseudoprime of four kinds.

Let $U_n$ be a nondegenerate Lucas sequence with parameters $P$ and $Q = ±1, \varepsilon = ±1$. Then, every arithmetic progression $ax + b$, where $(a,b) = 1$ which contains an odd integer $n_0$ with the Jacobi symbol $\left(\frac{D}{n_0}\right)$ equal to $\varepsilon$, contains infinitely many strong Lucas pseudoprimes $n$ with parameters $P$ and $Q = ±1$ such that $\left(\frac{D}{n} = \varepsilon \right)$ which are at the same time Lucas pseudoprimes of each of the four types.

A new identity is proved. From this we deduce as corollaries: 2.1–an identity involving the Lionville's function $\lambda(n)$, 2.2–an identity involving the von-Mongoldt's function $\lambda(n)$ and 2.3–an identity involving the generating function of Waring's problem. Some other identities are also mentioned.

We show a connection between the multidimensional Dirichlet, divisor problem and the zero free region for the Riemann zeta function.

This paper studies non-vanishing of quadratic twists of automorphic forms $f$ on $GL(2)$ over $\mathrm{Q}$ at various points inside the critical strip. Given any point $w_0$ inside the critical strip, and $\varepsilon > 0$, we show that at least $Y^{12/17-\varepsilon}$ of the quadratic twists $L(f,\chi_{d}, s)$ with $\vert d \vert \leq Y$ do not vanish inside the disc $\vert w - w_{0}\vert \lt (\mathrm{log}Y)^{-1-\varepsilon}$ (Here $d \equiv 1$ mod $4$ is a fundamental discriminant and $\chi_{d}$ denotes the Kronecker symbol.) If we assume the Ramanujan conjecture about the Fourier coefficients of $f$ (in particular, if $f$ is holomorphic) then $\frac{12}{17}$ above can be replaced with 1.

This should be compared with a result of Ono and Skinner [10] which states that if $f$ is a holomorphic newform of even weight and trivial character, then at least $\gg Y / \mathrm{log} Y$ of the quadratic twists $L(f,\chi_{d}, s)$ are nonzero at the central critical point. A slightly weaker result had been proved earlier by Perelli and Pomykala [11]. By contrast, we make no restriction on the holomorphy of $f$ and the result holds even if $f$ has non-trivial central character. Moreover, we prove non-vanishing in a disc about any point in the critical strip. As in [l1], our tools are the method of Iwaniec [4] and a mean value estimate of Heath-Brown [3].

In this paper we obtain a weighted form of Gonek's [5] uniform version of the classical Landau formula. The results are based on Kaczorowski-Perelli's [10] technique for the Riemann von Mangoldt explicit formula. The main feature of our results is a more flexible remainder term, which in particular gives better mean-values estimates.

The equation $m/(ax+b)=1/F_{1}(x)+1/F_{2}(x)+1/F_{3}(x)$ is shown to be impossible under some conditions on polynomials $ax+b$ and $F_{1}, \ F_{2}, \ F_{3}$.

In this paper we present a lower estimate on the number of non-zero solutions $(u,\mu)$ of the following boundary value problem $$\left\{ \begin{array}{l@{\quad \quad \quad}l} -\Delta u=\mu \cdot f(u) & \mathrm{on} & \Omega \\ u \equiv 0 & \mathrm{on} & \partial\Omega \end{array} \right. \qquad \qquad (\mathcal{P})$$ where $\mu \in \mathbb{R}, \Omega = (-\pi /2;\pi /2)^2$ and $f : \mathbb{R} \rightarrow \mathbb{R}$ is a function of class $C^1$ satisfying some additional requirements. By using the symmetry properties of the problem $(\mathcal{P})$ and classical results from number theory, we show that the numbers $\alpha_{\varepsilon}(L)$ of all distinct nontrivial solutions $(u, \mu)$ of $(\mathcal{P})$ such that $\|u\| \lt \varepsilon$, for $\varepsilon > 0$, where $0 \lt \mu \lt L + 1$, satisfy the following inequality $$\liminf_{\varepsilon \rightarrow 0} \ \alpha_{\varepsilon}(L) \geq \frac{5}{8} \pi L + O(\sqrt{L}) \quad as \ L \rightarrow \infty.$$

The rate of pointwise convergence of the Bézier type modification of some discrete Feller operators of locally bounded functions is estimated. In the general theorems the Chanturija modulus of variation is used. In particular, corresponding estimates for functions of bounded $p$-th power variation are deduced.

For a Borel measure $\mu$ on $[0, 1]$ with $1 \in \mathrm{supp}(\mu)$ we consider the Toeplitz operators $T_f$ with $f \in L^{\infty}(\mu \bigotimes \lambda)$ on the space $H^{2}(\mu)$ consisting of all holomorphic on $\mathbb{D}$ functions in the Lebegue class $L^{2}(\mu \bigotimes \lambda)$ on $\bar{\mathbb{D}}$. We show that for every Bergman-Hardy space $H^{2}(\mu)$ the following estimations hold: $\mathrm{dist}(T_{f}, \mathcal{K}(H^{2}(\mu))) \geqslant \vert f(t_0) \vert$ if $f$ is continuous at point $t_0 \in \mathbb{T}$, and $\mathrm{dist}(T_{f}, \mathcal{K}(H^{2}(\mu))) = \mathrm{sup}\{ \vert f(z) \vert : z \in \mathbb{D} \}$ if $f$ is a bounded harmonic function on $\mathbb{D}$.

We prove that there are infinitely many natural numbers $n$ for which the greatest common divisor of $2{n} - 3$ and $3^{n} - 2$ is divisible by $26665$.

A relationship between 3-free integers and zeros of the Riemann zeta-function, which is more explicit than the classical formula is presented.

In the present paper we derive an asymptotic formula for prime numbers $p$ for which $n$ exactly devides the order $a \ (\mathrm{mod} \ p)$.

We give an explicit description of Witt rings of of global function fields.