arXiv subjects

P. M. Gauthier

Publications and source records attributed to P. M. Gauthier.

10 recordsLinked to original sources

arXiv2024

Holomorphic motion, rational approximation and an equivalent formulation of the Riemann Hypothesis

A compact subset $K$ of the complex plane $\C$ is a set of polynomial (respectively rational) approximation if $P(K)=A(K)$ (respectively $R(K)=A(K)$), where $P(K)$ (respectively $R(K)$) is the family of functions on $K$ which are uniform limits of polynomials (respectively rational functions, having no poles on $K$) and $A(K)$ is the family of continuous functions on $K,$ which are holomorphic on the interior of $K.$ In the class of compact sets, the property of being a set of polynomial approximation is easily seen to be invariant under holomorphic motion. We show that this is no longer the case for rational approximation. Secondly, we show that the Riemann Hypothesis holds if and only if a certain map is a holomorphic motion.

math.CV↗

arXiv2018

Approximating the Riemann Zeta-function by Polynomials with Restricted Zeros

We approximate the Riemann Zeta-Function by polynomials and Dirichlet polynomials with restricted zeros.

math.CV↗

arXiv2014

Conformal extensions of functions defined on arbitrary subsets of Riemann surfaces

For a function defined on an arbitrary subset of a Riemann surface, we give conditions which allow the function to be extended conformally. One folkloric consequence is that two common definitions of an analytic arc in ${\mathbb C}$ are equivalent. Key-words: analytic continuation; analytic arc.

math.CV↗

arXiv2014

Density of polynomials in classes of functions on products of planar domains

We give sufficient conditions on planar domains for polynomials to be dense in the algebras A and A-infinity of the product of these domains, endowed with their natural topologies. We also characterize the uniform limits, with respect to the chordal metric, of polynomials on the product of the closures of these domains. The products may be finite products or infinite products, even uncountable.

math.CV↗

arXiv2013

Zero-free polynomial approximation on a chain of Jordan domains

On a compact subset of the plane with connected complement, is it possible to uniformly approximate a continuous function, holomorphic and non-vanishing on the interior, with polynomials non-vanishing on the entire compact set? In this brief note, we recall the surprising connection between this question and the Riemann hypothesis and proceed to provide an affirmative answer for a "chain" of Jordan domains.

math.CV↗

arXiv2010

On the Instability of the Riemann Hypothesis over Finite Fields

We show that it is possible to approximate the zeta-function of a curve over a finite field by meromorphic functions which satisfy the same functional equation and moreover satisfy (respectively do not satisfy) the analogue of the Riemann hypothesis. In the other direction, it is possible to approximate holomorphic functions by simple manipulations of such a zeta-function. We also consider the value distribution of zeta-functions of function fields over finite fields from the viewpoint of Nevanlinna theory.

math.CV↗

arXiv2009

Perturbations of L-functions with or without non-trivial zeros off the critical line

There exist small perturbations of L-functions, satisfying the appropriate functional equation, for which the analogue of the Riemann hypothesis fails radically. Moreover, this phenomenon is generic. However, there also exist small perturbations, for which the analogue of the Riemann hypothesis holds.

math.CV↗

arXiv2007

Rational Approximation for a Quasilinear Parabolic Equation

Approximation theorems, analogous to known results for linear elliptic equations, are obtained for solutions of the heat equation. Via the Cole-Hopf transformation, this gives rise to approximation theorems for a nonlinear parabolic equation, Burgers' equation.

math.AP↗

arXiv2006

Hartogs' theorem on separate holomorphicity for projective spaces

If a mapping of several complex variables into projective space is holomorphic in each pair of variables, then it is globally holomorphic.

math.CV↗

arXiv2006

Approximation on arcs and dendrites going to infinity in $C^n$ (extended version)

The Stone-Weierstrass approximation theorem is extended to certain unbounded sets in $C^n$. In particular, on a locally rectifiable arc going to infinity, each continuous function can be approximated by entire functions.

math.CV↗