Skip to main content Accessibility help
×
Home
Hostname: page-component-65dc7cd545-fz4lj Total loading time: 0.21 Render date: 2021-07-24T09:15:30.471Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

On the Size of an Expression in the Nyman–Beurling-Báez–Duarte Criterion for the Riemann Hypothesis

Published online by Cambridge University Press:  20 November 2018

Helmut Maier
Affiliation:
Department of Mathematics, University of Ulm, Helmholtzstrasse 18, 89081 Ulm, Germany, e-mail: helmut.maier@uni-ulm.de
Michael Th. Rassias
Affiliation:
Institute of Mathematics, University of Zurich, CH-8057, Zurich, Switzerland, e-mail : michail.rassias@math.uzh.ch
Corresponding

Abstract

A crucial role in the Nyman-Beurling-Báez-Duarte approach to the Riemann Hypothesis is played by the distance

$$d_{N}^{2}:=\underset{{{A}_{N}}}{\mathop{\inf }}\,\frac{1}{2\pi }\int _{-\infty }^{\infty }{{\left| 1-\zeta {{A}_{N}}\left( \frac{1}{2}+it \right) \right|}^{2}}\frac{dt}{\frac{1}{4}+{{t}^{2}}},$$

where the infimum is over all Dirichlet polynomials

$${{A}_{N}}\left( s \right)\,=\,\sum\limits_{n=1}^{N}{\frac{{{a}_{n}}}{{{n}^{s}}}}$$

of length $N$ . In this paper we investigate $d_{N}^{2}$ under the assumption that the Riemann zeta function has four nontrivial zeros off the critical line.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2018

Access options

Get access to the full version of this content by using one of the access options below.

References

[1] Bäez-Duarte, L., Balazard, M., Landreau, B., and Saias, E., Notes sur lafonction f de Riemann. III. Adv. Math. 149(2000), no. 1, 130144. http://dx.doi.Org/10.1006/aima.1999.1861Google Scholar
[2] Bäez-Duarte, L., Balazard, M., Landreau, B., and Saias, E., Etüde de Vautocorrelation multiplicative de lafonction ‘partiefractionnaire'. Ramanujan J. 9(2005), no. 1-2, 215240. http://dx.doi.Org/10.1007/s11139-005-0834-4Google Scholar
[3] Bettin, S., Conrey, J. B., and Farmer, D. W., An optimal choice of Dirichlet polynomiah for the Nyman-Beurling criterion. Proc. Steklov Inst. Math. 280(2013), suppl. 2, S30-S36. http://dx.doi.Org/10.1134/S0081543813030036Google Scholar
[4] Burnol, J. F., A lower bound in an approximation problem involving the zeros of the Riemann zeta function. Adv. Math. 170(2002), 56-70. http://dx.doi.Org/10.1006/aima.2001.2066Google Scholar
[5] Titchmarsh, E. C., The theory of the Riemann Zeta-function. Second ed., The Clarendon Press, Oxford University Press, New York, 1986.Google Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

On the Size of an Expression in the Nyman–Beurling-Báez–Duarte Criterion for the Riemann Hypothesis
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

On the Size of an Expression in the Nyman–Beurling-Báez–Duarte Criterion for the Riemann Hypothesis
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

On the Size of an Expression in the Nyman–Beurling-Báez–Duarte Criterion for the Riemann Hypothesis
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *