On generating functions in additive number theory, II: lower-order terms and applications to PDEs

We obtain asymptotics for sums of the form [Formula: see text] involving lower order main terms. As an application, we show that for almost all [Formula: see text] one has [Formula: see text] and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimensi...

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Autores principales: Brandes, J., Parsell, S. T., Poulias, C., Shakan, G., Vaughan, R. C.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer Berlin Heidelberg 2020
Materias:
Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7862536/
https://www.ncbi.nlm.nih.gov/pubmed/33603253
http://dx.doi.org/10.1007/s00208-020-02107-0
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author Brandes, J.
Parsell, S. T.
Poulias, C.
Shakan, G.
Vaughan, R. C.
author_facet Brandes, J.
Parsell, S. T.
Poulias, C.
Shakan, G.
Vaughan, R. C.
author_sort Brandes, J.
collection PubMed
description We obtain asymptotics for sums of the form [Formula: see text] involving lower order main terms. As an application, we show that for almost all [Formula: see text] one has [Formula: see text] and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrödinger and Airy equations.
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spelling pubmed-78625362021-02-16 On generating functions in additive number theory, II: lower-order terms and applications to PDEs Brandes, J. Parsell, S. T. Poulias, C. Shakan, G. Vaughan, R. C. Math Ann Article We obtain asymptotics for sums of the form [Formula: see text] involving lower order main terms. As an application, we show that for almost all [Formula: see text] one has [Formula: see text] and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrödinger and Airy equations. Springer Berlin Heidelberg 2020-12-23 2021 /pmc/articles/PMC7862536/ /pubmed/33603253 http://dx.doi.org/10.1007/s00208-020-02107-0 Text en © The Author(s) 2020 Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
spellingShingle Article
Brandes, J.
Parsell, S. T.
Poulias, C.
Shakan, G.
Vaughan, R. C.
On generating functions in additive number theory, II: lower-order terms and applications to PDEs
title On generating functions in additive number theory, II: lower-order terms and applications to PDEs
title_full On generating functions in additive number theory, II: lower-order terms and applications to PDEs
title_fullStr On generating functions in additive number theory, II: lower-order terms and applications to PDEs
title_full_unstemmed On generating functions in additive number theory, II: lower-order terms and applications to PDEs
title_short On generating functions in additive number theory, II: lower-order terms and applications to PDEs
title_sort on generating functions in additive number theory, ii: lower-order terms and applications to pdes
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7862536/
https://www.ncbi.nlm.nih.gov/pubmed/33603253
http://dx.doi.org/10.1007/s00208-020-02107-0
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