Resonance between the Representation Function and Exponential Functions over Arithemetic Progression
Let rn denote the number of representations of a positive integer n as a sum of two squares, i.e., n=x12+x22, where x1 and x2 are integers. We study the behavior of the exponential sum twisted by rn over the arithmetic progressions ∑n∼Xn≡lmodqrneαnβ, where 0≠α∈ℝ, 0<β<1, ex=e2πix, and n∼X means...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2021-01-01
|
| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2021/6616348 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850222240964018176 |
|---|---|
| author | Li Ma Xiaofei Yan |
| author_facet | Li Ma Xiaofei Yan |
| author_sort | Li Ma |
| collection | DOAJ |
| description | Let rn denote the number of representations of a positive integer n as a sum of two squares, i.e., n=x12+x22, where x1 and x2 are integers. We study the behavior of the exponential sum twisted by rn over the arithmetic progressions ∑n∼Xn≡lmodqrneαnβ, where 0≠α∈ℝ, 0<β<1, ex=e2πix, and n∼X means X<n≤2X. Here, X>1 is a large parameter, 1≤l≤q are integers, and l,q=1. We obtain the upper bounds in different situations. |
| format | Article |
| id | doaj-art-72ceac59e7ef4b36a7c6d25d0fe1f949 |
| institution | OA Journals |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-72ceac59e7ef4b36a7c6d25d0fe1f9492025-08-20T02:06:26ZengWileyJournal of Mathematics2314-46292314-47852021-01-01202110.1155/2021/66163486616348Resonance between the Representation Function and Exponential Functions over Arithemetic ProgressionLi Ma0Xiaofei Yan1School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong 250100, ChinaSchool of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong 250100, ChinaLet rn denote the number of representations of a positive integer n as a sum of two squares, i.e., n=x12+x22, where x1 and x2 are integers. We study the behavior of the exponential sum twisted by rn over the arithmetic progressions ∑n∼Xn≡lmodqrneαnβ, where 0≠α∈ℝ, 0<β<1, ex=e2πix, and n∼X means X<n≤2X. Here, X>1 is a large parameter, 1≤l≤q are integers, and l,q=1. We obtain the upper bounds in different situations.http://dx.doi.org/10.1155/2021/6616348 |
| spellingShingle | Li Ma Xiaofei Yan Resonance between the Representation Function and Exponential Functions over Arithemetic Progression Journal of Mathematics |
| title | Resonance between the Representation Function and Exponential Functions over Arithemetic Progression |
| title_full | Resonance between the Representation Function and Exponential Functions over Arithemetic Progression |
| title_fullStr | Resonance between the Representation Function and Exponential Functions over Arithemetic Progression |
| title_full_unstemmed | Resonance between the Representation Function and Exponential Functions over Arithemetic Progression |
| title_short | Resonance between the Representation Function and Exponential Functions over Arithemetic Progression |
| title_sort | resonance between the representation function and exponential functions over arithemetic progression |
| url | http://dx.doi.org/10.1155/2021/6616348 |
| work_keys_str_mv | AT lima resonancebetweentherepresentationfunctionandexponentialfunctionsoverarithemeticprogression AT xiaofeiyan resonancebetweentherepresentationfunctionandexponentialfunctionsoverarithemeticprogression |