Paul M. Gauthier

On Bloch’s constant

The well-known lower estimate for Bloch’s constantB is thatB > √3/4. Recently, M. Bonk has improved this estimate by showing thatB > √3/4 + 10-14. We give a further improvement by showing thatB ≥√3/4 + 2 × 10-14.

Approximation by harmonic functions on closed subsets of Riemann surfaces

Given continuous functionsu and ∈ on a closed subsetF of a Riemann surface, we seek a harmonic functionv on the surface (possibly with logarithmic singularities) such that |u−v|<∈ onF.

The Landau theorem and Bloch theorem for planar harmonic and pluriharmonic mappings

For a normalized quasiregular pluriharmonic mapping of the unit ball of into , we estimate the supremum of numbers such that some subdomain of the ball is mapped by diffeomorphically onto some ball of radius . Our estimates significantly improve earlier estimates, even in the case of harmonic functions in the disc.

Uniform approximation on closed sets by solutions of elliptic partial differential equations

Runges theorem asserts that each function holomorphic on a compact subset K of the complex plane C is the uniform limit of a sequence of rational functions {rn} and if C\K is connected we can take the rns to be polynomials. We generalize this result by replacing compact subsets by closed subsets and holomorphic functions by solutions of certain elliptic partial differential equations. The role of the rational functions will be played by solutions which are global except for isolated singularities of a certain kind.

Approximation of and by the Riemann Zeta-Function

It is possible to approximate the Riemann zeta-function by meromorphic functions which satisfy the same functional equation and satisfy (respectively do not satisfy) the analogue of the Riemann hypothesis.In the other direction, it is possible to approximate meromorphic functions by various manipulations of the Riemann zeta-function.

Universal overconvergence of homogeneous expansions of harmonic functions

On certain domains of Euclidean space, we establish the existence of harmonic functions, which are universal in the sense that their Taylor polynomials approximate all plausibly approximable functions in the complement of the domain.

Approximation by the Riemann zeta-function

Any meromorphic function having at most simple poles can be approximated by linear combinations of translates of the Riemann zeta-function. In particular, an arbitrary holomorphic function can be so approximated. If derivatives of the zeta-function are allowed, then arbitrary meromorphic functions can be approximated.

Small Perturbations of the Riemann Zeta Function and their Zeros

It is shown that the Riemann hypotheses fails for two types of arbitrarily close approximations of the Riemann zeta function. The first type is meromorphic and approximates the zeta function outside of a small set. The second type is quasi-meromorphic, agrees with the zeta function outside of an even smaller set and approximates the zeta function everywhere.

Harmonic approximation on closed subsets of Riemannian manifolds

We discuss the problem of approximating functions on a closed subset of a noncompact Riemannian manifold by functions which are harmonic on the entire manifold.

Approximating Functions by the Riemann Zeta-Function and by Polynomials with Zero Constraints

On certain compact sets K, we shall approximate functions having no zeros on the interior of K by translates of the Riemann zeta-function. As J. Andersson has shown recently, this is related to a natural problem in polynomial approximation.