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

Du, Yongguang, and Liu, Huaning. "On the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum." Czechoslovak Mathematical Journal 63.2 (2013): 461-473. <http://eudml.org/doc/260711>.

@article{Du2013,
abstract = {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 $k\ge \{1\}$ and $q\ge \{3\}$, and Dirichlet characters $\chi $, $\bar\{\chi \}$ modulo $q$ we define a mixed exponential sum \[ C(m,n;k;\chi ;\bar\{\chi \};q)= \sum \limits \_\{a=1\}^\{q\}\{\hspace\{-2.22214pt\}\vrule width0pt height1em\}^\{\prime \} \chi (a)G\_\{k\}(a,\bar\{\chi \})e \Big (\frac\{ma^\{k\}+n\overline\{a^\{k\}\}\}\{q\}\Big ), \] with Dirichlet character $\chi $ and general Gauss sum $G_\{k\}(a,\bar\{\chi \})$ as coefficient, where $\sum \nolimits ^\{\prime \}$ 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.},
author = {Du, Yongguang, Liu, Huaning},
journal = {Czechoslovak Mathematical Journal},
keywords = {mixed exponential sum; mean value; Dirichlet character; general Gauss sum; computational formula; Kloosterman sum; general Gauss sum; Dirichlet character},
language = {eng},
number = {2},
pages = {461-473},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum},
url = {http://eudml.org/doc/260711},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Du, Yongguang
AU - Liu, Huaning
TI - On the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 2
SP - 461
EP - 473
AB - 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 $k\ge {1}$ and $q\ge {3}$, and Dirichlet characters $\chi $, $\bar{\chi }$ modulo $q$ we define a mixed exponential sum \[ C(m,n;k;\chi ;\bar{\chi };q)= \sum \limits _{a=1}^{q}{\hspace{-2.22214pt}\vrule width0pt height1em}^{\prime } \chi (a)G_{k}(a,\bar{\chi })e \Big (\frac{ma^{k}+n\overline{a^{k}}}{q}\Big ), \] with Dirichlet character $\chi $ and general Gauss sum $G_{k}(a,\bar{\chi })$ as coefficient, where $\sum \nolimits ^{\prime }$ 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.
LA - eng
KW - mixed exponential sum; mean value; Dirichlet character; general Gauss sum; computational formula; Kloosterman sum; general Gauss sum; Dirichlet character
UR - http://eudml.org/doc/260711
ER -