We establish a connection between the L² norm of sums of dilated functions whose jth Fourier coefficients are $\left(j^{-\alpha }\right)$ for some α ∈ (1/2,1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L² and for the almost everywhere convergence of series of dilated functions.

We continue the investigation of convolutions of second order linear recursive sequences (see the first part in [1]). In this paper, we focus on the case when the characteristic polynomials of the sequences have common root.

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≢ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{{|}^p)}^{1/p}$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let ${\mu }_q(n,L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that $\left|Q\left(0\right)\right|>1/L({\sum }_{j=1}^n{\left|Q\left(j\right)\right|}^q{)}^{1/q}$. We find the size of ${\kappa }_p(n,L)$ and ${\mu }_q(n,L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about ${\mu }_{\infty }(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even in that special...

We use the estimation of the number of integers $n$ such that $\lfloor n^c\rfloor$ belongs to an arithmetic progression to study the coprimality of integers in ${\mathbb{N}}^c={\left\{\lfloor n^c\rfloor \right\}}_{n\in \mathbb{N}}$, $c>1$, $c\notin \mathbb{N}$.

Let $M$ be a family of Mumford-type, that is, a family of polarized complex abelian fourfolds as introduced by Mumford in [9]. This family is defined starting from a quaternion algebra $A$ over a real cubic number field and imposing a condition to the corestriction of such $A$. In this paper, under some extra conditions on the algebra $A$, we make this condition explicit and in this way we are able to describe the polarization and the complex structures of the fibers. Then, we look at the non simple $CM$-fibers...