Peter Borwein

Polynomials and Polynomial Inequalities

Chaptern 1 Introduction and Basic Properties.- 2 Some Special Polynomials.- 3 Chebyshev and Descartes Systems.- 4 Denseness Questions.- 5 Basic Inequalities.- 6 Inequalities in Muntz Spaces.- Inequalities for Rational Function Spaces.- Appendix A1 Algorithms and Computational Concerns.- Appendix A2 Orthogonality and Irrationality.- Appendix A3 An Interpolation Theorem.- Appendix A5 Inequalities for Polynomials with Constraints.- Notation.

On the rapid computation of various polylogarithmic constants

We give algorithms for the computation of the d-th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of log(2) or π on a modest work station in a few hours run time.

A cubic counterpart of Jacobi's identity and the AGM

We produce exact cubic analogues of Jacobis celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is

The Riemann Hypothesis

a_n+1 := a_n + 2b_n / 3

Computational Excursions in Analysis and Number Theory

and b_n+1 := [formula cannot be replicated]. The limit of this iteration is identified in terms of the hypergeometric function ₂F₁ (1/3, 2/3; 1 ; ·), which supports a particularly simple cubic transformation.

Some cubic modular identities of Ramanujan

The german mathematician Bernhard Riemann only had a short life, nevertheless he contributed challenging new ideas and concepts to mathematics. His invention of topological methods in complex analysis and his foundation of Riemannian geometry made him one of the most influential mathematicians of his time. In addition he worked on differential geometry, differential equations, and mathematical physics. His one and only article [43] on number theory, entitled ’On the number of

Ramanujan, modular equations, and approximations to Pi or how to compute one billion digits of Pi

Preface.- Introduction.- LLL and PSLQ.- Pisot and Salem Numbers.- Rudin-Shapiro Polynomials.- Fekete Polynomials.- Products of Cyclotomic Polynomials.- Location of Zeros.- Maximal Vanishing.- Diophantine Approximation of Zeros.- The Integer-Chebyshev Problem.- The Prouhet-Tarry-Escott Problem.- The Easier Waring Problem.- The Erdos-Szekeres Problem.- Barker Polynomials and Golay Pairs.- The Littlewood Problem.- Spectra.- Appendix A: A Compendium of Inequalities.- B: Lattice Basis Reduction and Integer Relations.- C: Explicit Merit Factor Formulae.- D: Research Problems.- References.- Index.

A complete description of golay pairs for lengths up to 100

There is a beautiful cubic analogue of Jacobis fundamental theta function identity: θ⁴₃ = θ⁴₄ + θ⁴₂. It is

A survey of Sylvester's problem and its generalizations

(\sum_{n,m=-\infty}^{\infty} q^{n^2+nm+m^2})³ = (\sum_{n,m=-\infty}^{\infty} ω^{n-m}q^{n²+nm+m²})³ + (\sum_{n,m=-\infty}^{\infty} q^{(n+1/3)²+(n+1/3)(m+1/3)+(m+1/3)²})³.

The quest for PI

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