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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.-… (More)

We produce exact cubic analogues of Jacobi's celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is $a_n+1 := a_n + 2b_n… (More)

- Peter Borwein
- 2002

Preface.- Introduction.- LLL and PSLQ.- Pisot and Salem Numbers.- Rudin-Shapiro Polynomials.- Fekete Polynomials.- Products of Cyclotomic Polynomials.- Location of Zeros.- Maximal Vanishing.-… (More)

The Muntz-Legendre polynomials arise by orthogonalizing the Muntz system {xIo, x'l~, . .. } with respect to Lebesgue measure on [0, 1] . In this paper, differential and integral recurrence formulae… (More)

- Peter Borwein
- 1991

We prove that if q is an integer greater than one and r is a non-zero rational ( r ≠ −q m ) then Σ n =1 ∞ (1/( q n + r )) is irrational and is not a Liouville number.

- Peter Borwein
- 1995

A very simple class of algorithms for the computation of the Riemann-zeta function to arbitrary precision in arbitrary domains is proposed. These algorithms out perform the standard methods based on… (More)

- Peter Borwein
- 1992

We prove that the series are irrational and not Liouville whenever q is an integer ( q ╪ 0, ±1) and r is a nonzero rational ( r ╪ − q n ).

to the Riemann Hypothesis.- Why This Book.- Analytic Preliminaries.- Algorithms for Calculating ?(s).- Empirical Evidence.- Equivalent Statements.- Extensions of the Riemann Hypothesis.- Assuming the… (More)

G. Giuga conjectured that if an integer n satisfies \sum\limits_{k=1}^{n-1} k^{n-1} \equiv -1 mod n, then n must be a prime. We survey what is known about this interesting and now fairly old… (More)

- Peter Borwein
- 2002

A classical problem in Diophantine equations that occurs in many guises is the Prouhet-Tarry-Escott problem. This is the problem of finding two distinct lists (repeats are allowed) of integers [α… (More)