# On a kind of generalized Lehmer problem

Ontology type: schema:ScholarlyArticle      Open Access: True

### Article Info

DATE

2012-12

AUTHORS ABSTRACT

For 1 ⩾ c ⩾ p − 1, let E1,E2, …,Em be fixed numbers of the set {0, 1}, and let a1, a2, …, am (1 ⩽ ai ⩽ p, i = 1, 2, …,m) be of opposite parity with E1,E2, …,Em respectively such that a1a2…am ≡ c (mod p). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N(c,m,p) = {1 \over {{2^{m - 1}}}}\mathop {\sum\limits_{{a_1} = 1}^{p - 1} {\sum\limits_{{a_2} = 1}^{p - 1} \ldots } }\limits_{{a_1}{a_2} \ldots \equiv c{\rm{ (}}\bmod {\rm{ }}p)} \sum\limits_{{a_m} = 1}^{p - 1} {(1 - {{( - 1)}^{{a_1} + {E_1}}})(1 - {{( - 1)}^{{a_2} + {E_2}}}) \ldots } (1 - {( - 1)^{{a_m} + {E_m}}}).$$\end{document} We are interested in the mean value of the sums \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\limits_{c = 1}^{p - 1} {{E^2}} (c,m,p),$$\end{document} where E(c, m, p) = N(c,m, p)−((p − 1)m−1)/(2m−1) for the odd prime p and any integers m ⩾ 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. More... »

PAGES

1135-1146

### Journal

TITLE

Czechoslovak Mathematical Journal

ISSUE

4

VOLUME

62

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s10587-012-0068-8

DOI

http://dx.doi.org/10.1007/s10587-012-0068-8

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1043218980

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