zbMATH — the first resource for mathematics

Scientific computation on mathematical problems and conjectures. (English) Zbl 0703.65004
CBMS-NSF Regional Conference Series in Applied Mathematics, 60. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. vi, 122 p. $ 21.50 (1990).
The purpose of these lecture notes is to study the use of scientific computation as a tool in attacking a number of mathematical problems and conjectures. In Chapter 1, the Bernstein conjecture of 1913 in polynomial approximation theory is studied, which was recently settled (negatively). In Chapter 2, a “1/9” conjecture of 1977 in rational approximation theory is examined. High-precision calculations give strong indication that this conjecture is also false. In Chapter 3, a survey is presented related to the famous Riemann hypothesis of 1859, and a weaker form known as the Pólya conjecture of determining lower bounds for the de Bruijn- Newman constant \(\Lambda\). In Chapter 4, the asymptomatic behavior of the zeros of the partial sums of exp(z) is analyzed. In Chapter 5, real and complex best rational approximations are compared. The main result of Chapter 6 is a generalization of Jensen’s inequality for polynomials having concentration of low degrees is discussed.
The emphasis is this lecture note rests strongly on the interface between hard analysis and high-precision calculations. Each chapter has its own list of references.
Reviewer: F.Szidarovszky

65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
41-02 Research exposition (monographs, survey articles) pertaining to approximations and expansions
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
41A20 Approximation by rational functions
65H05 Numerical computation of solutions to single equations
30C20 Conformal mappings of special domains