In this article, we consider the expressions for content-parametrized Schur multiple zeta-functions in terms of multiple zeta-functions of Euler-Zagier type and their star-variants, or in terms of modified zeta-functions of root systems. First of all, we focus on the Schur multiple zeta-function of hook type. And then, applying the Giambelli formula and induction argument, we obtain the expressions for general content-parametrized Schur multiple zeta-functions.

We survey the potential for progress in additive number theory arising from recent advances concerning major arc bounds associated with mean value estimates for smooth Weyl sums. We focus attention on the problem of representing large positive integers as sums of a square and a number of $k$-th powers. We show that such representations exist when the number of\break $k$-th powers is at least $\lfloor c_0 k\rfloor +2$, where $c_0=2.13629\ldots$. By developing an abstract framework capable of handling sequences with appropriate distribution properties, analogous conclusions are obtained, for example, when the square is restricted to have prime argument.

We present a streamlined account of a recent theorem on the classification of the $L$-functions of degree $2$ and conductor $1$ from the extended Selberg class. We also present a more general new result dealing with functional equations involving two Dirichlet series. Further, we correct a slip in the original proof of the above theorem, which however does not affect the final result.

We evaluate friable averages of arithmetic functions whose Dirichlet series is analytically close to some complex power of the Riemann zeta function. We obtain asymptotic expansions resembling those provided by the Selberg-Delange method in the non-friable case. Some applications are provided to the friable distribution of the additive function counting the total number of prime factors.

Analysis of special Dirichlet polynomials and estimates for Kloosterman sums are described which may lead to improvements of the classical zero-density theorems for the Riemann zeta function.

We study “forbidden” conductors, i.e. numbers $q>0$ satisfying algebraic criteria introduced by J. Kaczorowski, A. Perelli and M. Radziejewski, that cannot be conductors of $L$-functions of degree $2$ from the extended Selberg class. We show that the set of forbidden $q$ is dense in the interval $(0,4)$, solving a problem posed in [6]. We also find positive points of accumulation of rational forbidden $q$.

Assuming the generalized Lindelöf hypothesis, we provide asymptotic formulas for the mean values of the first and second moments of Ramanujan sums over any number field. Additionally, unconditionally, we estimate the second moment of Ramanujan sums over cyclotomic number fields.

Some methods for testing the reducibility of matrix continued fractions(MCFs) will be presented. The methods introduced in this paper areeffective (i.e. the result is obtained in a~finite number of steps).We will consider applications of some reducibility methods for analysingmatrix continued fractions and matrix Padé approximants.