This monograph uses results from polynomial interpolation, approximation theory and orthogonal polynomials to obtain asymptotic formulae for the Riemann zeta function \(\zeta(s)\) and the Dirichlet zeta function \(\beta(s)\), that extend the only known facts for fixed \(s\) \[ \zeta(s)=\sum_{n=1}^N\,n^{-s}+o(1),\;\beta(s)=\sum_{n=1}^N\,(-1)^n (2n+1)^{-s}+o(1), \] as \(N\rightarrow\infty\) and \(\text{Re}\,s>1\) for \(\zeta(s)\) and \(\text{Re}\,s>0\) for \(\beta(s)\).
The booklet has the following layout
§1. Introduction (3 pages)
Contains the layout of the monograph and some historical remarks.
§2. Integral formulae for the interpolation error-term (20 pages)
An extension of explicit formulae for the interpolation error to Cauchy-type integrals is given for a fairly general set of interpolation points (which further on usually are taken to be real). Interpolation formulae are given for \(|y|^s(\text{sgn}\,y)^{\ell} \log^{\nu}|y|\), where \(s\in\mathbb{C},\;\ell=0\text{ or }1\text{ and }\nu\) is a nonnegative integer.
§3. Asymptotic properties of special sequences of polynomials (36 pages)
Three classes are studied; the polynomials are subject to conditions on the zeros, coefficients, asymptotics, normalization.
§4. Pointwise asymptotic relations between the interpolation error and zeta functions (20 pages)
Apart form the relations, some asymptotic formulae are given using a.o. normalized Chebyshev polynomials of the first and the second kind, normalized polynomials with equidistant zeroes, normalized Williams-Apostol polynomials of the first and second kind.
§5. Asymptotic relations between the interpolation error and zeta functions (20 pages)
§6. Other applications (8 pages)
This covers applications to universal exponential sums, functional equations for zeta functions and to combinatorial representations for Bernoulli and Euler numbers.
References
The list contains 57 items.