Search

Start Over Subject asymptotic formula

1. A hybrid mean value involving two-term exponential sums and polynomial character sums

Creator:
Di, Han
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Dirichlet character of polynomials, two-term exponential sums, hybrid mean value, and asymptotic formula
Language:
English
Description:
Let $q \ge 3$ be a positive integer. For any integers $m$ and $n$, the two-term exponential sum $C(m,n,k;q)$ is defined by $C(m,n,k;q) = \sum _{a=1}^q e ({(ma^k +na)}/{q})$, where $e(y)={\rm e}^{2\pi {\rm i} y}$. In this paper, we use the properties of Gauss sums and the estimate for Dirichlet character of polynomials to study the mean value problem involving two-term exponential sums and Dirichlet character of polynomials, and give an interesting asymptotic formula for it.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. A hybrid mean value related to certain Hardy sums and Kloosterman sums

Creator:
Guo, Xiaoyan and Zhang, Wenpeng
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Hardy sums, the Kloosterman sums, hybrid mean value, asymptotic formula, and identity
Language:
English
Description:
The main purpose of this paper is using the mean value formula of Dirichlet L-functions and the analytic methods to study a hybrid mean value problem related to certain Hardy sums and Kloosterman sums, and give some interesting mean value formulae and identities for it.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

4. 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

5. On Lehmer's problem and Dedekind sums

Creator:
Pan, Xiaowei and Zhang, Wenpeng
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Lehmer's problem, error term, Dedekind sums, hybrid mean value, and asymptotic formula
Language:
English
Description:
Let $p$ be an odd prime and $c$ a fixed integer with $(c, p)=1$. For each integer $a$ with $1\le a \leq p-1$, it is clear that there exists one and only one $b$ with $0\leq b \leq p-1$ such that $ab \equiv c $ (mod $p$). Let $N(c, p)$ denote the number of all solutions of the congruence equation $ab \equiv c$ (mod $p$) for $1 \le a$, $b \leq p-1$ in which $a$ and $\overline {b}$ are of opposite parity, where $\overline {b}$ is defined by the congruence equation $b\overline {b}\equiv 1\pmod p$. The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet $L$-functions to study the hybrid mean value problem involving $N(c,p)-\frac {1}{2}\phi (p)$ and the Dedekind sums $S(c,p)$, and to establish a sharp asymptotic formula for it.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

6. On the $2k$-th power mean of $\frac {L'}L(1,\chi )$ with the weight of Gauss sums

Creator:
Ren, Dongmei and Yi, Juan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Dirichlet L-function, Gauss sums, and asymptotic formula
Language:
English
Description:
The main purpose of this paper is to study the hybrid mean value of $\frac {L'}L(1,\chi )$ and Gauss sums by using the estimates for trigonometric sums as well as the analytic method. An asymptotic formula for the hybrid mean value $\sum _{\chi \neq \chi _0} |\tau (\chi )| |\frac {L'}L(1,\chi )|^{2k}$ of $\frac {L'}L$ and Gauss sums will be proved using analytic methods and estimates for trigonometric sums.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

7. On the mean value of a sum analogous to character sums over short intervals

Creator:
Ganglian, Ren and Wenpeng, Zhang
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
character sums, short intervals, even power mean, and asymptotic formula
Language:
English
Description:
The main purpose of this paper is to study the mean value properties of a sum analogous to character sums over short intervals by using the mean value theorems for the Dirichlet L-functions, and to give some interesting asymptotic formulae.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

8. On the mean value of the generalized Dirichlet $L$-functions

Creator:
Ma, Rong, Yi, Yuan, and Zhang, Yulong
Type:
model:article and TEXT
Subject:
generalized Dirichlet $L$-functions, mean value properties, functional equation, and asymptotic formula
Language:
English
Description:
Let $q\ge 3$ be an integer, let $\chi $ denote a Dirichlet character modulo $q.$ For any real number $a\ge 0$ we define the generalized Dirichlet $L$-functions $$ L(s,\chi ,a)=\sum _{n=1}^{\infty }\frac {\chi (n)}{(n+a)^s}, $$ where $s=\sigma +{\rm i} t$ with $\sigma >1$ and $t$ both real. They can be extended to all $s$ by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet $L$-functions especially for $s=1$ and $s=\frac 12+{\rm i} t$, and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

9. 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

10. Some new sums related to D. H. Lehmer problem

Creator:
Zhang, Han and Zhang, Wenpeng
Format:
print, bez média, and svazek
Type:
model:article and TEXT
Subject:
Lehmer number, analytic method, trigonometric sums, asymptotic formula, 13, and 51
Language:
English
Description:
About Lehmer’s number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let p be a prime, and let N(k; p) denote the number of all 1\leqslant a_{i}\leq p-1 such that a_{1}a_{2}...a_{k}\equiv 1 mod p and 2 | ai + āi + 1, i = 1, 2, ..., k. The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function N(k; p), and give an interesting asymptotic formula for it., Han Zhang, Wenpeng Zhang., and Obsahuje seznam literatury
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public