| Title:
|
Two identities related to Dirichlet character of polynomials (English) |
| Author:
|
Yao, Weili |
| Author:
|
Zhang, Wenpeng |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
1 |
| Year:
|
2013 |
| Pages:
|
281-288 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $q$ be a positive integer, $\chi $ denote any Dirichlet character $\mod q$. For any integer $m$ with $(m, q)=1$, we define a sum $C(\chi, k, m; q)$ analogous to high-dimensional Kloosterman sums as follows: $$ C(\chi, k, m; q)=\sum _{a_1=1}^{q}{}' \sum _{a_2=1}^{q}{}' \cdots \sum _{a_k=1}^{q}{}' \chi (a_1+a_2+\cdots +a_k+m\overline {a_1a_2\cdots a_k}), $$ where $a\cdot \overline {a}\equiv 1\bmod q$. The main purpose of this paper is to use elementary methods and properties of Gauss sums to study the computational problem of the absolute value $|C(\chi, k, m; q)|$, and give two interesting identities for it. (English) |
| Keyword:
|
Dirichlet character of polynomials |
| Keyword:
|
sum analogous to Kloosterman sum |
| Keyword:
|
identity |
| Keyword:
|
Gauss sum |
| MSC:
|
11L05 |
| MSC:
|
11L40 |
| idZBL:
|
Zbl 1274.11126 |
| idMR:
|
MR3035511 |
| DOI:
|
10.1007/s10587-013-0018-0 |
| . |
| Date available:
|
2013-03-01T16:21:45Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143184 |
| . |
| Reference:
|
[1] Burgess, D. A.: On Dirichlet characters of polynomials.Proc. Lond. Math. Soc., III. Ser. 13 (1963), 537-548. Zbl 0118.04704, MR 0148627, 10.1112/plms/s3-13.1.537 |
| Reference:
|
[2] Granville, A., Soundararajan, K.: Large character sums: pretentious characters and the Pólya-Vinogradov theorem.J. Am. Math. Soc. 20 (2007), 357-384. Zbl 1210.11090, MR 2276774, 10.1090/S0894-0347-06-00536-4 |
| Reference:
|
[3] Smith, R. A.: On $n$-dimensional Kloosterman sums.J. Number Theory 11 (1979), 324-343. Zbl 0409.10024, MR 0544261, 10.1016/0022-314X(79)90006-4 |
| Reference:
|
[4] Ye, Y.: Estimation of exponential sums of polynomials of higher degrees. II.Acta Arith. 93 (2000), 221-235. Zbl 0953.11028, MR 1759916, 10.4064/aa-93-3-221-235 |
| Reference:
|
[5] Zhang, W., Yi, Y.: On Dirichlet characters of polynomials.Bull. Lond. Math. Soc. 34 (2002), 469-473. Zbl 1038.11052, MR 1897426, 10.1112/S0024609302001030 |
| Reference:
|
[6] Zhang, W., Yao, W.: A note on the Dirichlet characters of polynomials.Acta Arith. 115 (2004), 225-229. Zbl 1076.11048, MR 2100501, 10.4064/aa115-3-3 |
| . |
