On the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum

Title:

On the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum

Creator:

Du, Yongguang and Liu, Huaning

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:3782a7c2-2c40-494b-8515-2a379b9631be
uuid:3782a7c2-2c40-494b-8515-2a379b9631be

Subject:

mixed exponential sum, mean value, Dirichlet character, general Gauss sum, and computational formula

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

The main purpose of the paper is to study, using the analytic method and the property of the Ramanujan's sum, the computational problem of the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum. For integers $m$, $ n$, $ k$, $ q$, with $k\geq {1}$ and $q\geq {3}$, and Dirichlet characters $\chi $, $\bar {\chi }$ modulo $q$ we define a mixed exponential sum $$ C(m,n;k;\chi ;\bar {\chi };q)= \sum \limits _{a=1}^{q}{\mkern -4mu\vrule width0pt height1em}' \chi (a)G_{k}(a,\bar {\chi })e \Big (\frac {ma^{k}+n\overline {a^{k}}}{q}\Big ), $$ with Dirichlet character $\chi $ and general Gauss sum $G_{k}(a,\bar {\chi })$ as coefficient, where $\sum \nolimits '$ denotes the summation over all $a$ such that $(a,q)=1$, $a\bar {a}\equiv {1}\mod {q}$ and $e(y)={\rm e}^{2\pi {\rm i} y}$. We mean value of $$ \sum _{m}\sum _{\chi }\sum _{\bar {\chi }}|C(m,n;k;\chi ;\bar {\chi };q)|^{4}, $$ and give an exact computational formula for it.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

461-473

Source:

Czechoslovak Mathematical Journal | 2013 Volume:63 | Number:2

Harvested from:

CDK

Metadata only:

false