| Title:
|
On Lehmer's problem and Dedekind sums (English) |
| Author:
|
Pan, Xiaowei |
| Author:
|
Zhang, Wenpeng |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
61 |
| Issue:
|
4 |
| Year:
|
2011 |
| Pages:
|
909-916 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Lehmer's problem |
| Keyword:
|
error term |
| Keyword:
|
Dedekind sums |
| Keyword:
|
hybrid mean value |
| Keyword:
|
asymptotic formula |
| MSC:
|
11F20 |
| MSC:
|
11L40 |
| MSC:
|
11M20 |
| idZBL:
|
Zbl 1249.11090 |
| idMR:
|
MR2886246 |
| DOI:
|
10.1007/s10587-011-0058-2 |
| . |
| Date available:
|
2011-12-16T15:36:18Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141796 |
| . |
| 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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| Reference:
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| Reference:
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