Geodesic
Approximation in measure: the Dirichlet problem, universality and~the~Riemann hypothesis
Izvestiya. Mathematics , Tome 85 (2021) no. 3, pp. 547-561.

Voir la notice de l'article provenant de la source Math-Net.Ru

We use approximation in measure to solve an asymptotic Dirichlet problem on arbitrary open sets and to show that many functions, including the Riemann zeta-function, are universal in measure. Connections with the Riemann hypothesis are suggested.
Keywords: harmonic approximation in measure, harmonic, holomorphic, Dirichlet problem, Riemann zeta-function, universality. 
@article{IM2_2021_85_3_a14,
     author = {J. Falc\'o and P. M. Gauthier},
     title = {Approximation in measure: the {Dirichlet} problem, universality {and~the~Riemann} hypothesis},
     journal = {Izvestiya. Mathematics },
     pages = {547--561},
     publisher = {mathdoc},
     volume = {85},
     number = {3},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2021_85_3_a14/}
}
TY  - JOUR
AU  - J. Falcó
AU  - P. M. Gauthier
TI  - Approximation in measure: the Dirichlet problem, universality and~the~Riemann hypothesis
JO  - Izvestiya. Mathematics 
PY  - 2021
SP  - 547
EP  - 561
VL  - 85
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2021_85_3_a14/
LA  - en
ID  - IM2_2021_85_3_a14
ER  - 
%0 Journal Article
%A J. Falcó
%A P. M. Gauthier
%T Approximation in measure: the Dirichlet problem, universality and~the~Riemann hypothesis
%J Izvestiya. Mathematics 
%D 2021
%P 547-561
%V 85
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2021_85_3_a14/
%G en
%F IM2_2021_85_3_a14
J. Falcó; P. M. Gauthier. Approximation in measure: the Dirichlet problem, universality and~the~Riemann hypothesis. Izvestiya. Mathematics , Tome 85 (2021) no. 3, pp. 547-561. http://geodesic.mathdoc.fr/item/IM2_2021_85_3_a14/

[1] K.-G. Grosse-Erdmann, “Universal families and hypercyclic operators”, Bull. Amer. Math. Soc. (N.S.), 36:3 (1999), 345–381 | DOI | MR | Zbl

[2] P. M. Gauthier, F. Sharifi, “Luzin-type holomorphic approximation on closed subsets of open Riemann surfaces”, Canad. Math. Bull., 60:2 (2017), 300–308 | DOI | MR | Zbl

[3] P. M. Gauthier, F. Sharifi, “Luzin-type harmonic approximation on subsets of non-compact Riemannian manifolds”, J. Math. Anal. Appl., 474:2 (2019), 1132–1152 | DOI | MR | Zbl

[4] J. Falcó, P. M. Gauthier, “An asymptotic holomorphic boundary problem on arbitrary open sets in Riemann surfaces”, J. Approx. Theory, 257 (2020), 105451, 11 pp. | DOI | MR | Zbl

[5] M. Craioveanu, M. Puta, T. M. Rassias, “Canonical differential operators associated to a Riemannian manifold”, Old and new aspects in spectral geometry, Ch. 2, Math. Appl., 534, Kluwer Acad. Publ., Dordrecht, 2001, 75–117 | DOI | MR | Zbl

[6] J. Andersson, “Mergelyan's approximation theorem with nonvanishing polynomials and universality of zeta-functions”, J. Approx. Theory, 167 (2013), 201–210 | DOI | MR | Zbl

[7] J. Steuding, Value-distribution of $L$-functions, Lecture Notes in Math., 1877, Springer, Berlin, 2007, xiv+317 pp. | DOI | MR | Zbl

[8] T. Bagby, P. M. Gauthier, J. Woodworth, “Tangential harmonic approximation on Riemannian manifolds”, Harmonic analysis and number theory (Montreal, PQ, 1996), CMS Conf. Proc., 21, Amer. Math. Soc., Providence, RI, 1997, 58–72 | MR | Zbl

[9] D. H. Armitage, P. M. Gauthier, “Recent developments in harmonic approximation, with applications”, Results Math., 29:1-2 (1996), 1–15 | DOI | MR | Zbl

[10] K. Matsumoto, “A survey on the theory of universality for zeta and $L$-functions”, Number theory. Plowing and starring through high wave forms (Fukuoka, 2013), Ser. Number Theory Appl., 11, World Sci. Publ., Hackensack, NJ, 2015, 95–144 | DOI | MR | Zbl

[11] E. C. Titchmarsh, The theory of the Riemann zeta-function, Edited and with a preface by D. R. Heath-Brown, 2nd ed., The Clarendon Press, Oxford Univ. Press, New York, 1986, x+412 pp. | MR | Zbl

[12] B. Bagchi, “Recurrence in topological dynamics and the Riemann hypothesis”, Acta Math. Hungar., 50:3-4 (1987), 227–240 | DOI | MR | Zbl