This article is dedicated to Lithuanian number theorist Professor Antanas Laurinˇcikas on
the occasion of his 70th birthday. We sketch the main stages in the development of his scientific
career. Although A. Laurinˇcikas started with probabilistic number theory, later on he became
one of the leading world scientists in the theory of zeta-functions, especially concerning their
universality. In the review we give a brief account of his pre-university life and the development
of his career as a mathematician from the time he entered Vilnius University. We review some
results of Antanas starting with early ones and then higlight the main results. At the end a list
of scientific publications of A. Laurinˇcikas is presented.

The article is dedicated to the memory of George Voronoi. It  is concerned with (p-adic) L-functions (in partially  (p-adic) zeta functions)  and cyclotomic  (p-adic) (multiple) zeta values. The beginning of the article contains a short  summary of the results on the Bernoulli numbers associated with the studies of George Voronoi. Results on multiple zeta values have presented by D. Zagier, by P. Deligne and A.Goncharov, by A. Goncharov, by F. Brown, by C. Glanois and others. S. "Unver have investigated p-adic multiple zeta values in the depth two. Tannakian interpretation of p-adic multiple zeta values is given by H.  Furusho. Short history and connections among Galois groups, fundamental groups, motives and arithmetic functions are presented in the talk by Y. Ihara. Results  on multiple zeta values, Galois groups and geometry of modular varieties has presented by Goncharov.Interesting unipotent motivic fundamental group is defined and investigated by Deligne and Goncharov. The framework of (p-adic) L-functions and (p-adic) (multiple) zeta values is based on Kubota-Leopoldt p-adic L-functions and arithmetic p-adic L-functions by Iwasawa. Motives and  (p-adic) (multiple) zeta values by Glanois and by "Unver, improper intersections of Kudla-Rapoport divisors and Eisenstein series by Sankaran are reviewed. More fully the content of the article can be found at the following table of contents: Introduction. 1. Voronoi-type congruences for  Bernoulli numbers. 2. Riemann zeta values. 3. On class groups of rings with divisor theory.  Imaginary quadratic and cyclotomic fields. 4. Eisenstein Series. 5. Class group, class fields and zeta functions. 6. Multiple zeta values. 7. Elements of non-Archimedean local fields and $ p-$adic analysis. 8. Iterated integrals and (multiple) zeta values. 9. Formal groups and p-divisible groups. 10. Motives and (p-adic) (multiple) zeta values. 11. On the Eisenstein series associated with Shimura varieties. Sections 1-9 and subsection 11.1 (On some Shimura varieties and Siegel modular forms) can be considered as an elementary introduction to the results of section 10 and subsection 11.2 (On improper intersections of Kudla-Rapoport divisors and Eisenstein series). Numerical examples are included.

For the classical Riesz potential or the fractional integral $$I_{\alpha}$$, the Hardy--Littlewood-- Sobolev--Stein--Weiss $$(L^p, L^q)$$-boundedness conditions with power weights are well known. Using the Fourier transform $$\mathcal{F}$$, the Riesz potential is determined by the equality $$\mathcal{F}(I_{\alpha}f)(y)=|y|^{-\alpha}\mathcal{F}(f)(y)$$. An important generalization of the Fourier transform became the Dunkl transform $$\mathcal{F}_k(f)$$, acting in Lebesgue spaces with Dunkl's weight, defined by the root system $$R\subset \mathbb{R}^d$$, its reflection group G and a non-negative multiplicity function k on R, invariant with respect to G.S. Thangavelu and Yu.~Xu using the equality $$\mathcal{F}_k (I_{\alpha}^kf)(y)=|y|^{-\alpha}\mathcal{F}_k(f)(y)$$ determined the D-Riesz potential $$I_{\alpha}^k$$. For the D-Riesz potential, the boundedness conditions in Lebesgue spaces with Dunkl weight and power weights, similar to the conditions for the Riesz potential, were also proved. At the conference "Follow-up Approximation Theory and Function Spaces"   in the Centre de Recerca Matem`atica (CRM, Barcelona, 2017) M.L. Goldman raised the question about $$(L_p,L_q) $$-boundedness conditions of the D-Riesz potential with piecewise-power weights. Consideration of piecewise-power weights makes it possible to reveal the influence of the behavior of weights at zero and infinity on the boundedness of the D-Riesz potential. This paper provides a complete answer to this question. In particular, in the case of the Riesz potential, necessary and sufficient conditions are obtained. As auxiliary results, necessary and sufficient conditions for the boundedness of the Hardy and Bellman operators are proved in Lebesgue spaces with Dunkl weight and piecewise-power weights.

The paper studies the Zeta function $$\zeta(M(p_1,p_2)|\alpha)$$ of the monoid $$M (p_1,p_2)$$ generated by Prime numbers $$p_1<p_2$$ of the form 3n+2. Next,the main monoid $$M_{3,1}(p_1,p_2)\subset M(p_1,p_2)$$ and the main set $$ A_{3,1}(p_1,p_2)= M(p_1,p_2)\setminus M_{3,1}(p_1, p_2)$$ are distinguished. For the corresponding Zeta functions, explicit finite formulas are found that give an analytic continuation on the entire complex plane except for the countable set of poles. Inverse series for these Zeta functions and functional equations are found.
The paper gives definitions of three new types of monoids of natural numbers with a unique decomposition into simple elements: monoids of degrees, Euler monoids modulo q and unit monoids modulo q. Provided the expression of the Zeta functions using the Euler product.
The paper discusses the effect Davenport-Heilbronn Zeta-functions of monoids of natural numbers that is associated with the appearance of zeros of the Zeta-functions of terms obtained by the classes of residues modulo.
For monoids with an exponential sequence of primes, the barrier series hypothesis is proved and it is shown that the holomorphic domain of the Zeta function of such a monoid is the complex half-plane to the right of the imaginary axis.
In conclusion, topical problems with zeta-functions of monoids of natural numbers that require further investigation are considered.

For every monoid M of natural numbers defined a new class of periodic functions $$M_s^\alpha$$, which is a subclass of a known class of periodic functions Korobov $$E_s^\alpha$$. With respect to the norm $$\|f(\vec{x})\|_{E_s^\alpha}$$, the class $$M_s^\alpha$$ is an inseparable Banach subspace of class $$E_s^\alpha$$.
It is established that the class $$M_s^\alpha$$ is closed with respect to the action of the Fredholm integral operator and the Fredholm integral equation of the second kind is solvable on this class.
In this paper we obtain estimates of the image norm of the integral operator, which contain the kernel norm and the s-th degree of the Zeta function of the monoid M. Estimates are obtained for the parameter $$\lambda$$, in which the integral operator $$A_{\lambda,f}$$ is a compression. The theorem on the representation of the unique solution of Fredholm integral equation of the second kind in the form of Neumann series is proved.
The paper deals with the problems of solving the partial differential equation with the differential operator $$Q\left(\frac{\partial }{\partial x_1},\ldots,\frac{\partial }{\partial x_s}\right)$$ in the space $$M^\alpha_{s}$$, which depends on the arithmetic properties of the spectrum of this operator.
A paradoxical fact is found that for a monoid $$M_{q,1}$$ of numbers comparable to 1 modulo q, a quadrature formula with a parallelepiped grid for an admissible set of coefficients modulo q is exact on the class $$M_{q,1,s}^\alpha$$. Moreover, this statement remains true for the class $$M_{q,a,s}^\alpha$$ with 1 < a < q when q is a Prime number. Since the functions of class $$M_{q,a,s}^\alpha$$ with 1 < a < q do not have a zero Fourier coefficient $$C(\vec{0})$$, then for a simple q the sum of the function values at the nodes of the corresponding parallelepipedal grid will be zero.

In this paper, for an arbitrary monoid of natural numbers, the foundations of the Dirichlet series algebra are constructed either over a numerical field or over a ring of integers of an algebraic numerical field.
For any numerical field $$\mathbb{K}$$, it is shown that the set $$\mathbb{D}^*(M)_{\mathbb{K}}$$ of all reversible Dirichlet series of $$\mathbb{D}(M)_{\mathbb{K}}$$ is an infinite Abelian group consisting of series whose first coefficient is nonzero.
We introduce the notion of an integer Dirichlet monoid of natural numbers that form an algebra over a ring of algebraic integers $$\mathbb{Z}_\mathbb{K}$$ of the algebraic field $$\mathbb{K}$$. It is shown that for a group $$\mathbb{U}_\mathbb{K}$$ of algebraic units of the ring of algebraic integers of $$\mathbb{Z}_\mathbb{K}$$ an algebraic field $$\mathbb{K}$$ the set of $$\mathbb{D}(M)_{\mathbb{U}_\mathbb{K}}$$ of entire Dirichlet series, $$a(1)\in\mathbb{U}_\mathbb{K}$$, is multiplicative group.
For any Dirichlet series from the Dirichlet series algebra of a monoid of natural numbers, the reduced series, the irreversible part and the additional series are determined. A formula for decomposition of an arbitrary Dirichlet series into the product of the reduced series and a construction of an irreversible part and an additional series is found.
For any monoid of natural numbers allocated to the algebra of Dirichlet series, convergent in the entire complex domain. The Dirichlet series algebra with a given half-plane of absolute convergence is also constructed. It is shown that for any nontrivial monoid M and for any real $$\sigma_0$$, there is an infinite set of Dirichlet series of $$\mathbb{D}(M)$$ such that the domain of their holomorphism is $$\alpha$$-half-plane $$\sigma>\sigma_0$$.
With the help of the universality theorem S. M. Voronin managed to prove the weak form of the universality theorem for a wide class of Zeta functions of monoids of natural numbers.
In conclusion describes the actual problem with the Zeta functions of monoids of natural numbers that require further research. In particular, if the Linnik-Ibrahimov hypothesis is true, then a strong theorem of universality should be valid for them.

In this note we give a necessary and sufficient condition on the triplet of nonnegative integers a < b < c for which the Newman polynomial $$\sum_{j=0}^a x^j + \sum_{j=b}^c x^j$$ has a root on the unit circle. From this condition we derive that for each $$d \geq 3$$ there is a positive integer $$n>d$$ such that the Newman polynomial $$1+x+\dots+x^{d-2}+x^n$$ of length d has no roots on the unit circle.

An abelian group G is called a TI-group if every associative ring with additive group G is filial. An abelian group G such that every (associative) ring with additive group G is an SI-ring (a hamiltonian ring) is called an SI-group (an $$SI_H$$-group). In this paper, TI-groups, as well as SI-groups and $$SI_H$$-groups are described in the class of reduced algebraically compact abelian groups.

A multiplication on an abelian group G is a homomorphism $$\mu: G\otimes G\rightarrow G$$. An mixed abelian group G is called an MT-group if every multiplication on the torsion part of the group G can be extended  uniquely to a multiplication on G. MT-groups have been studied in many articles on the theory of additive groups of rings, but their complete description has not yet been obtained. In this paper, a pure fully invariant subgroup $$G^*_\Lambda$$ is considered for an abelian MT-group G. One of the main properties of this subgroup is that $$\bigcap\limits_{p \in \Lambda (G)}pG^*_\Lambda$$ is a nil-ideal in every ring with the additive group G (here $$\Lambda (G)$$ is the set of all primes p, for which the p-primary component of G is non-zero). It is shown that for every MT-group G either $$G=G^*_\Lambda$$ or the quotient group $$G/G^*_\Lambda$$ is uncountable.

Let $$x_0,x_1,...$$ be a sequence of points in $$[0,1)^s$$. A subset $$S$$ of $$[0,1)^s$$ is called a bounded remainder set if there exist two real numbers a and C such that, for every integer N, %We say that $$S \subset [0,1)^s$$ is a bounded remainder set with respect to the sequence $$(x_n)_{n \geq 1}$$ if there is a constant C such that $$ | {\rm card}\{n <N \; | \; x_{n} \in S\} - a N| <C . $$ Let $$ (x_n)_{n \geq 0} $$ be an s-dimensional Halton-type sequence obtained from a global function field, $$b \geq 2$$, $$\gamma =(\gamma_1,...,\gamma_s)$$, $$\gamma_i \in [0, 1)$$, with $$b$$-adic expansion $$\gamma_i= \gamma_{i,1}b^{-1}+ \gamma_{i,2}b^{-2}+...$$, $$i=1,...,s$$. In this paper, we prove that $$[0,\gamma_1) \times ...\times [0,\gamma_s)$$ is the bounded remainder set with respect to the sequence $$(x_n)_{n \geq 0}$$ if and only if \begin{equation} \nonumber \max_{1 \leq i \leq s} \max \{ j \geq 1 \; | \; \gamma_{i,j} \neq 0 \} < \infty. \end{equation}
We also obtain the similar results for a generalized Niederreiter sequences, Xing-Niederreiter sequences and Niederreiter-Xing sequences.

The periodicity and quasi-periodicity of functional continued fractions in the hyperelliptic field $$L = \mathbb{Q}(x)(\sqrt {f})$$ has a more complex nature, than the periodicity of the numerical continued fractions of the elements of a quadratic fields. It is known that the periodicity of a continued fraction of the element $$\sqrt{f}/h^{g + 1}$$, constructed by valuation associated with a polynomial h of first degree, is equivalent to the existence of nontrivial S-units in a field L of the genus g and is equivalent to the existence nontrivial torsion in a group of classes of divisors. This article has found an exact interval of values of $$s \in \mathbb{Z}$$ such that the elements $$\sqrt {f}/h^s $$ have a periodic decomposition into a continued fraction, where $$f \in \mathbb{Q}[x] $$ is a squarefree polynomial of even degree. For polynomials f of odd degree, the problem of periodicity of continued fractions of elements of the form $$\sqrt {f}/h^s $$ are discussed in the article [5], and it is proved that the length of the quasi-period does not exceed degree of the fundamental S-unit of L. The problem of periodicity of continued fractions of elements of the form $$\sqrt {f}/h^s$$ for polynomials f of even degree is more complicated. This is underlined by the example we found of a polynomial f of degree 4, for which the corresponding continued fractions have an abnormally large period length. Earlier in the article [5] we found examples of continued fractions of elements of the hyperelliptic field L with a quasi-period length significantly exceeding the degree of the fundamental S-unit of L.

Probabilistic methods are used in the theory of zeta-functions since Bohr and Jessen time (1910-1935). In 1930, they proved the first theorem for the Riemann zeta-function $$\zeta(s)$$, $$s=\sigma+it$$, which is a prototype of modern limit theorems characterizing the behavior of $$\zeta(s)$$ by weakly convergent probability measures. More precisely, they obtained that, for $$\sigma>1$$, there exists the limit $$\lim_{T\to\infty} \frac{1}{T} \mathrm{J} \left\{t\in[0,T]: \log\zeta(\sigma+it)\in R\right\}, $$ where R is a rectangle on the complex plane with edges parallel to the axes, and $$\mathrm{J}A$$ denotes the Jordan measure of a set $$A\subset \mathbb{R}$$. Two years latter, they extended the above result to the half-plane $$\sigma>\frac{1}{2}$$. Ideas of Bohr and Jessen were developed by Wintner, Borchsenius, Jessen, Selberg and other famous mathematicians. Modern versions of the Bohr-Jessen theorems, for a wide class of zeta-functions, were obtained in the works of K. Matsumoto. The theory of Bohr and Jessen is applicable, in general, for zeta-functions having Euler's product over primes. In the present paper, a limit theorem for a zeta-function without Euler's product is proved. This zeta-function is a generalization of the classical Hurwitz zeta-function. Let $$\alpha$$, $$0<\alpha \leqslant 1$$, be a fixed parameter, and $$\mathfrak{a}=\{a_m: m\in \mathbb{N}_0= \mathbb{N}\cup\{0\}\}$$ be a periodic sequence of complex numbers. The periodic Hurwitz zeta-function $$\zeta(s,\alpha; \mathfrak{a})$$ is defined, for $$\sigma>1$$, by the Dirichlet series $$\zeta(s,\alpha; \mathfrak{a})=\sum_{m=0}^\infty \frac{a_m}{(m+\alpha)^s}, $$ and is meromorphically continued to the whole complex plane. Let $$\mathcal{B}(\mathbb{C})$$ denote the Borel $$\sigma$$-field of the set of complex numbers, $$\mathrm{meas}A$$ be the Lebesgue measure of a measurable set $$A\subset \mathbb{R}$$, and let the function $$\varphi(t)$$ for $$t\geqslant T_0$$ have the monotone positive derivative $$\varphi'(t)$$ such that $$(\varphi'(t))^{-1}=o(t)$$ and $$\varphi(2t) \max_{t\leqslant u\leqslant 2t} (\varphi'(u))^{-1}\ll t$$. Then it is obtained in the paper that, for $$\sigma>\frac{1}{2}$$, $$\frac{1}{T} \mathrm{meas}\left\{t\in[0,T]: \zeta(\sigma+i\varphi(t), \alpha; \mathfrak{a})\in A\right\},\quad A\in \mathcal{B}(\mathbb{C}), $$ converges weakly to a certain explicitly given probability measure on $$(\mathbb{C}, \mathcal{B}(\mathbb{C}))$$ as $$T\to\infty$$.

In paper the  problem of sound wave scattering by absolutely rigid cylinder with radially inhomogeneous isotropic elastic coating in a planar waveguide is considered. It is believed that a waveguide filled with a homogeneous ideal fluid, one of its borders is absolutely rigid and the other --- acoustically soft, heterogeneity laws of a coating material are described by differentiable functions, harmonic sound wave excited by a given distribution of sources in the section waveguide. In the case of steady state oscillations the propagation of small perturbations in ideal fluid is described by the  Helmholtz's equation. The oscillations of an inhomogeneous isotropic elastic cylindrical layer described by general motion equations  of the continuous medium. The boundary-value problem for the system of ordinary second order differential equations  is constructed for determination of the displacement field in inhomogeneous coating. The primary field of disturbances is represented by a set of its own waveguide waves. The pressure of the field scattered by the cylindrical body is sought as potential of a simple layer. The Green function for the Helmholtz equation that satisfies the given boundary conditions on the waveguide walls and conditions of radiation at infinity is constructed.  The function of distribution density of sources are sought as a Fourier series expansion. The infinite linear system of equations is obtained for determination of the coefficients of this decomposition. The solution of truncated infinite system  is  found by the inverse matrix method. Analytical expressions for the scattered acoustic field  in different areas of the waveguide are obtained.

The conception of Generalized Gaussian Sum $$G_f(m)$$ for  a periodic arithmetical functon with a period, is equal prime number q, for integers m,n is introduce: $$ G_f(m)=\sum_{n=1}^{q-1}\left(\frac nq\right)f\left(\frac{mn}q\right). $$ Here are considered the particular cases $$f(x)=B_\nu(\{x\}), \nu\geq 1,$$ where $$B_\nu(x)$$ - Bernoulli polynomials. The paper uses the technique of finite Fourier series. If the function $$f\left(\frac{k}{q}\right)$$ is defined at $$k=0,1,\ldots,q-1,$$ it can be decomposed into a finite Fourier series $$ f\left(\frac{k}{q}\right)=\sum_{m=0}^{q-1}c_me^{2\pi i\frac{mk}{q}}, \quad c_m=\frac{1}{q}\sum_{k=0}^{q-1}f\left(\frac{k}{q}\right)e^{-2\pi i\frac{mk}{q}}. $$ By decomposition into a finite Fourier series of a generalized Gauss sum $$ G_\nu(m)=G_\nu(m;B_\nu)=\sum_{n=1}^{q-1}\left(\frac nq\right)B_\nu{\left(\left\{x+\frac{mn}q\right\}\right)} $$ for $$\nu=1$$ and $$\nu=2$$ , new formulas are found that Express the value of the Legendre symbol through the full sums of periodic functions. This circumstance makes it possible to obtain new analytical properties of the corresponding Dirichlet series and arithmetic functions, which will be the topic of the following works. An important property of the sums $$G_1$$ and $$G_2,$$ namely: $$G_1\ne 0,$$ if $$q\equiv 3\pmod 4$$ and $$G_1=0,$$ if $$q\equiv 1\pmod 4;$$ $$G_2= 0,$$ if $$q\equiv 3\pmod 4$$ and $$G_2=\frac 1{q^2}\sum\limits_{n=1}^{q-1}n^2\left(\frac nq\right),$$ if $$q\equiv 1\pmod 4.$$