arXiv subjects

Xavier Ros-Oton

Publications and source records attributed to Xavier Ros-Oton.

74 records · Page 5Linked to original sources

arXiv2012

On a factorization of Riemann's $ζ$ function with respect to a quadratic field and its computation

Let $K$ be a quadratic field, and let $ζ_K$ its Dedekind zeta function. In this paper we introduce a factorization of $ζ_K$ into two functions, $L_1$ and $L_2$, defined as partial Euler products of $ζ_K$, which lead to a factorization of Riemann's $ζ$ function into two functions, $p_1$ and $p_2$. We prove that these functions satisfy a functional equation which has a unique solution, and we give series of very fast convergence to them. Moreover, when $Δ_K>0$ the general term of these series at even positive integers is calculated explicitly in terms of generalized Bernoulli numbers.

math.NT↗

arXiv2011

Periodic solutions with nonconstant sign in Abel equations of the second kind

The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation $x\mapsto x^{-1}$, and there are many results available in the literature for the first kind equation. However, the equivalence breaks down when one seeks for solutions with nonconstant sign. This note is devoted to periodic solutions with nonconstant sign in Abel equations of the second kind. Specifically, we obtain sufficient conditions to ensure the existence of a periodic solution that shares the zeros of the leading coefficient of the Abel equation. Uniqueness and stability features of such solutions are also studied.

math.CA↗