Polynomial interpolation and asymptotic representations for zeta functions
Volume 496 / 2013
Abstract
We develop various asymptotic relations between the Riemann zeta function $\zeta(s)$ and the interpolation errors of Lagrange and Hermite interpolation to functions like $|y|^s$ and $y^{2m}\log |y|$. We show that the interpolation nodes of these interpolation processes include zeros of Gegenbauer and Hermite polynomials and polynomials with equidistant zeros. Similar results are valid for the Dirichlet beta function $\beta(s)$ as well. So the results of the monograph serve as the bridge between the theory of zeta functions and polynomial interpolation, one of the most studied areas of analysis.
Several applications of major asymptotics to properties of zeta functions are presented. In particular, we develop new criteria for $\zeta(s)=0$ and $\beta(s)=0$ in the critical strip. Other applications include construction of universal exponential sums (in the spirit of Voronin's universality theorem), limit summary formulae for $\zeta(s)$ and $\beta(s)$, and new combinatorial representations for Bernoulli and Euler numbers.
