In this paper we study Dirichlet convolution with a given arithmetical function $f$ as a~linear mapping $\varphi_f$ that sends a sequence $(a_n)$ to $(b_n)$ where $b_n=\sum_{d|n} f(d)a_{n/d}$. We investigate when this is a bounded operator on $l^2$ and find the operator norm. Of particular interest is the case $f(n)=n^{-\alpha}$ for its connection to the Riemann zeta function on the line $\Re s =\alpha$. For $\alpha>1$, $\varphi_f$ is bounded with $\|\varphi_f\|=\zeta(\alpha)$. For the unbounded case, we show that $\varphi_f:\mathcal{M}^2\to\mathcal{M}^2$ where $\mathcal{M}^2$ is the subset of $l^2$ of multiplicative sequences, for many $f\in\mathcal{M}^2$. Consequently, we study the `quasi'-norm \[ \sup_{\substack{\|a\|=T\\ a\in\mathcal{M}^2} } \frac{\|\varphi_f a\|}{\|a\|} \] for large $T$, which measures the `size' of $\varphi_f$ on $\mathcal{M}^2$. For the $f(n)=n^{-\alpha}$ case, we show this quasi-norm has a striking resemblance to the conjectured maximal order of $|\zeta (\alpha+iT)|$ for $\alpha>\frac{1}{2}$.

In this paper we generalize some results, obtained by Shimura, on the critical values of $L$-functions of $l$-adic representations attached to quadratic CM-base change of Hilbert modular forms twisted by finite order characters, to the case of the critical values of $L$-functions of arbitrary base change to CM-number fields of $l$-adic representations attached to Hilbert modular forms twisted by some finite-dimensional representations.

We prove that the sum $\sum_{\left\{d\le x, (d,r)=1\right.}\mu(d)/d^{1+\varepsilon}$ is bounded by $1+\varepsilon$, uniformly in $x\ge1$, $r$ and $\varepsilon>0$. We prove a similar estimate for the quantity $\sum_{\left\{d\le x, (d,r)=1\right.}\mu(d)\log(x/d)/d^{1+\varepsilon}$. When $\varepsilon=0$, $r$ varies between 1 and a hundred, and $x$ is below a million, this sum is non-negative and this raises the question as to whether it is non-negative for every~$x$.

We obtain, by means of the technique introduced in by J. Kaczorowski, a meromorphic continuation and the functional equation for the function $m(F, \cdot)$, where $F$ is from the Selberg class with a functional equation expressible with exactly one $\Gamma$ function.

In this paper we study the four-dimensional divisor problem of $(a,b,c,c)$ type, where $1\leq a\leq b<c$ are fixed integers. Our theorems improve some classical results. As an application, we study the error term of the summatory function of the exponential totient function.

In this paper, we use Ramanujan's Schläfli-type modular equations to find some new values of class invariants $g_n$ and also give alternate proofs of some of known values.

Bowman and Bradley proved an explicit formula for the sum of multiple zeta values whose indices are the sequence $(3,1,3,1,\ldots,3,1)$ with a number of $2$'s inserted. Kondo, Saito and Tanaka considered the similar sum of multiple zeta-star values and showed that this value is a rational multiple of a power of $\pi$. In this paper, we give an explicit formula for the rational part. In addition, we interpret the result as an identity in the harmonic algebra.

Given an integer $q\ge 2$, a $q$-normal number is an irrational number $\eta$ such that any preassigned sequence of $k$ digits occurs in the $q$-ary expansion of $\eta$ at the expected frequency, namely $1/q^k$. Given an integer $q\ge 3$, we consider the sequence of primes reduced modulo $q$ and examine various possibilities of constructing normal numbers using this sequence. We create a sequence of independent random variables that mimics the sequence of primes and then show that for almost all outcomes this allows to obtain a normal number.

Absolute continuity of an embedding between Banach function spaces is an interesting property which is closely related to compactness. In this paper we study absolutely continuous embeddings between arbitrary Banach spaces intermediate with respect to the couple $(L_{1}(\Omega), L_{\infty}(\Omega))$. Our results allow to check if an embedding of such spaces is absolutely continuous. Applications related with the degree of proximity between two function spaces are established for the case $\Omega=[0,1]$ and $\Omega=[0,\infty)$.

In this paper, we study the local points of small degrees on Shimura curves $X_0^D(N)$ over a totally real field $F$. We then study the $\bar\mathbb{F}_p$-gonality for these Shimura curves in the case $F=\mathbb{Q}$.

We give a result on the asymptotic behavior of multiple zeta-functions near non-positive integers. By using that result, we evaluate limit values of multiple zeta-functions at non-positive integers.

For $k \geq 2$ we bound the number of pairs of consecutive $k$-full numbers.

Let $\varphi= \{\varphi_m\}_{m=1}^{\infty}$ be a family of convex functions $\varphi_m$ on ${\mathbb R}^n$ with certain growth conditions. With a help of restrictions of functions $\varphi_m$ on ${\mathbb Z}^n$ a weighted space of functions on ${\mathbb Z}^n$ denoted as $A_{\varphi}$ is defined. Linear continuous functionals on this space in terms of their Fourier-Laplace transform are described. This description and functional analysis methods allowed to study surjectivity of difference operators on $A_{\varphi}$ and spectral synthesis problem in the kernel of such operators for a special case of a family $\varphi$.

In this paper, we study the structure of $End(M)$ where $M$ is a motive of $GL_2$-type over a totally real field. Under some standard conjectures for motives, we then prove some results about Galois representations attached to such motives.

It is proved that there are Dedekind's zeta functions with multiple zeros inthe critical strip of arbitrarily large multiplicity. There are given examplesof such zeros of the Dedekind zeta function of the field $\mathbb{Q}(\zeta_3, {\root 3 \of 5})$.

We find all possible cycle-lengths of polynomial mappings in one variable over rings of integers of number fields of signature $(0,2)$. Such fields have unit rank $1$, and possible cycle-lengths for other fields having unit rank $\le 1 $, but other signature, were found earlier by other authors.