dc.creator | Arias de Reyna Martínez, Juan | es |
dc.date.accessioned | 2016-06-29T07:44:32Z | |
dc.date.available | 2016-06-29T07:44:32Z | |
dc.date.issued | 2015-08 | |
dc.identifier.citation | Arias de Reyna Martínez, J. (2015). On the distribution (mod 1) of the normalized zeros of the Riemann Zeta-function. Journal of Number Theory, 153, 37-53. | |
dc.identifier.issn | 1096-1658 | es |
dc.identifier.uri | http://hdl.handle.net/11441/42875 | |
dc.description.abstract | We consider the problem whether the ordinates of the non-trivial zeros of ζ(s)
are uniformly distributed modulo the Gram points, or equivalently, if the normalized zeros
(xn) are uniformly distributed modulo 1. Odlyzko conjectured this to be true. This is far
from being proved, even assuming the Riemann hypothesis (RH, for short).
Applying the Piatetski-Shapiro 11/12 Theorem we are able to show that, for 0 < κ < 6/5,
the mean value 1
N
P
n≤N exp(2πiκxn) tends to zero. The case κ = 1 is especially interesting.
In this case the Prime Number Theorem is sufficient to prove that the mean value is 0, but
the rate of convergence is slower than for other values of κ. Also the case κ = 1 seems to
contradict the behavior of the first two million zeros of ζ(s).
We make an effort not to use the RH. So our Theorems are absolute. We also put forward
the interesting question: will the uniform distribution of the normalized zeros be compatible
with the GUE hypothesis?
Let ρ =
1
2 + iα run through the complex zeros of zeta. We do not assume the RH so that
α may be complex. For 0 < κ < 6
5 we prove that
lim
T→∞
1
N(T)
X
0<Re α≤T
e
2iκϑ(α) = 0
where ϑ(t) is the phase of ζ(
1
2 + it) = e
−iϑ(t)Z(t). | es |
dc.description.sponsorship | Ministerio de Economía y Competitividad | es |
dc.format | application/pdf | es |
dc.language.iso | eng | es |
dc.publisher | Elsevier | es |
dc.relation.ispartof | Journal of Number Theory, 153, 37-53. | |
dc.rights | Attribution-NonCommercial-NoDerivatives 4.0 Internacional | * |
dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/4.0/ | * |
dc.subject | zeta function | es |
dc.subject | zeros of zeta | es |
dc.subject | equidistribution | es |
dc.subject | GUE hypothesis | es |
dc.subject | normalized zeros | es |
dc.title | On the distribution (mod 1) of the normalized zeros of the Riemann Zeta-function | es |
dc.type | info:eu-repo/semantics/article | es |
dcterms.identifier | https://ror.org/03yxnpp24 | |
dc.type.version | info:eu-repo/semantics/submittedVersion | es |
dc.rights.accessRights | info:eu-repo/semantics/openAccess | es |
dc.contributor.affiliation | Universidad de Sevilla. Departamento de Análisis Matemático | es |
dc.relation.projectID | info:eu-repo/grantAgreement/MINECO/MTM2012-30748 | es |
dc.identifier.doi | http://dx.doi.org/10.1016/j.jnt.2015.01.006 | es |
idus.format.extent | 14 p. | es |
dc.journaltitle | Journal of Number Theory | es |
dc.publication.volumen | 153 | es |
dc.publication.initialPage | 37 | es |
dc.publication.endPage | 53 | es |
dc.identifier.idus | https://idus.us.es/xmlui/handle/11441/42875 | |
dc.contributor.funder | Ministerio de Economía y Competitividad (MINECO). España | |