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Start Over Subject Dirichlet $L$-function

1. A hybrid mean value of the Dedekind sums

Creator:
Yang, Hai
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Dedekind sums, Dirichlet $L$-function, and mean value
Language:
English
Description:
The main purpose of this paper is to use the M. Toyoizumi's important work, the properties of the Dedekind sums and the estimates for character sums to study a hybrid mean value of the Dedekind sums, and give a sharper asymptotic formula for it.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. On a kind of generalized Lehmer problem

Creator:
Rong and Zhang, Yulong
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Lehmer problem, character sum, Dirichlet $L$-function, and asymptotic formula
Language:
English
Description:
For $1\le c\le p-1$, let $E_1,E_2,\dots ,E_m$ be fixed numbers of the set $\{0,1\}$, and let $a_1, a_2,\dots , a_m$ $(1\le a_i\le p$, $i=1,2,\dots , m)$ be of opposite parity with $E_1,E_2,\dots ,E_m$ respectively such that $a_1a_2\dots a_m\equiv c\pmod p$. Let \begin {equation*} N(c,m,p)=\frac {1}{2^{m-1}}\mathop {\mathop {\sum }_{a_1=1}^{p-1} \mathop {\sum }_{a_2=1}^{p-1}\dots \mathop {\sum }_{a_m=1}^{p-1}} _{a_1a_2\dots a_m\equiv c\pmod p} (1-(-1)^{a_1+E_1})(1-(-1)^{a_2+E_2})\dots (1-(-1)^{a_m+E_m}). \end {equation*} \endgraf We are interested in the mean value of the sums \begin {equation*} \sum _{c=1}^{p-1}E^2(c,m,p), \end {equation*} where $ E(c,m,p)=N(c,m,p)-({(p-1)^{m-1}})/({2^{m-1}})$ for the odd prime $p$ and any integers $m\ge 2$. When $m=2$, $c=1$, it is the Lehmer problem. In this paper, we generalize the Lehmer problem and use analytic method to give an interesting asymptotic formula of the generalized Lehmer problem.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

3. Second moments of Dirichlet $L$-functions weighted by Kloosterman sums

Creator:
Wang, Tingting
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
general $k$-th Kloosterman sum, Dirichlet $L$-function, the mean square value, and asymptotic formula
Language:
English
Description:
For the general modulo $q\geq 3$ and a general multiplicative character $\chi $ modulo $q$, the upper bound estimate of $ |S(m, n, 1, \chi , q)| $ is a very complex and difficult problem. In most cases, the Weil type bound for $ |S(m, n, 1, \chi , q)| $ is valid, but there are some counterexamples. Although the value distribution of $ |S(m, n, 1, \chi , q)| $ is very complicated, it also exhibits many good distribution properties in some number theory problems. The main purpose of this paper is using the estimate for $k$-th Kloosterman sums and analytic method to study the asymptotic properties of the mean square value of Dirichlet $L$-functions weighted by Kloosterman sums, and give an interesting mean value formula for it, which extends the result in reference of W. Zhang, Y. Yi, X. He: On the $2k$-th power mean of Dirichlet L-functions with the weight of general Kloosterman sums, Journal of Number Theory, 84 (2000), 199–213.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public