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pborwein (AT) cecm.sfu.ca

ASB 10905

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Peter Borwein

Research Interests

My research lives at the nexus of classical analysis and number theory. The tools are analytic, probabilistic, combinatorial and number theoretic with a central computational flavour. The computational techniques are sophisticated, require significant computational resources (hundreds of hours of grid computing in some instances) and the algorithms themselves are sometimes the central issue (as with the Merit Factor Problem).

Substantial symbolic and numeric calculation has led to the discovery of some rather beautiful analytic objects (series, iterations, etc.), and in many cases, such as the derivation of Ramanujan type series considerably aids the proofs. The questions that arise impinge on issues in approximation theory, number theory and computational complexity (not to mention the obvious Computational issues).

This research has also led to some of the most efficient known algorithms for various elementary functions and constants (a number of the recent record calculations of pi have used one of our algorithms). A pleasant by-product has been the detection of various phenomena of a kind probably not visible without substantial interactive computation.

Specific Areas
  • Expansion problems of a number theoretic variety. The relationship between orthogonal expansions and irrationality proofs (such as zeta(3)). Growth problems concerning polynomials with integer coefficients associated with conjectures of Erdos and Szekeres.
  • Location of zeros of special polynomials and functions. Like the Riemann Zeta function.
  • Iterations and expansions related to special functions, emphasizing low complexity expansions.
  • Analytic inequalities and expansions for rational functions and polynomials, including lacunary polynomials.
  • Approximation issues concerning Chebyshev systems, their Chebyshev polynomials, Muntz systems and incomplete rationals. The related conjectures of D.J. Newman.
  • Specific examples of problems of interest include the following "old plums" from diophantine and combinatorial number theory. These problems have all been open for half a century.

  • Lehmer's Problem
  • Show that any monic polynomial with integer coefficients that is irreducible and is not a cyclotomic polynomial has Mahler measure at least 1.1762 ...

  • Mahler's Problem
  • For each n, find the polynomial of degreen and height 1 that have largest possible Mahler measure.

  • Conjecture of Schinzel and Zassenhaus
  • There is a constant c>0 such that any monic polynomial p of degree n with integer coefficients either has Mahler measure 1 or has at least one root of modulus at least 1+c/n$.

  • The Merit Factor Problem of Golay and Littlewood
  • Find a polynomial of degree n with coefficients +1 and -1 only that has smallest possible L_4 norm on the unit disk. Or smallest possibe supremum norm.