| Title:
|
On a kind of generalized Lehmer problem (English) |
| Author:
|
Ma, Rong |
| Author:
|
Zhang, Yulong |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
4 |
| Year:
|
2012 |
| Pages:
|
1135-1146 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Lehmer problem |
| Keyword:
|
character sum |
| Keyword:
|
Dirichlet $L$-function |
| Keyword:
|
asymptotic formula |
| MSC:
|
11A25 |
| MSC:
|
11M06 |
| MSC:
|
11N37 |
| idZBL:
|
Zbl 1259.11090 |
| idMR:
|
MR3010261 |
| DOI:
|
10.1007/s10587-012-0068-8 |
| . |
| Date available:
|
2012-11-10T21:51:48Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143049 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| . |
