On a kind of generalized Lehmer problem View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2012-12

AUTHORS

Rong Ma, Yulong Zhang

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

References to SciGraph publications

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


Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

[
  {
    "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
    "about": [
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Mathematical Sciences", 
        "type": "DefinedTerm"
      }, 
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Pure Mathematics", 
        "type": "DefinedTerm"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "alternateName": "School of Science, Northwestern Polytechnical University, 710072, Xi\u2019an, Shaanxi, P.R.China", 
          "id": "http://www.grid.ac/institutes/grid.440588.5", 
          "name": [
            "School of Science, Northwestern Polytechnical University, 710072, Xi\u2019an, Shaanxi, P.R.China"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Ma", 
        "givenName": "Rong", 
        "id": "sg:person.014071323365.09", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014071323365.09"
        ], 
        "type": "Person"
      }, 
      {
        "affiliation": {
          "alternateName": "The School of Electronic and Information Engineering, Xi\u2019an Jiaotong University, 710049, Xi\u2019an, Shaanxi, P.R.China", 
          "id": "http://www.grid.ac/institutes/grid.43169.39", 
          "name": [
            "The School of Electronic and Information Engineering, Xi\u2019an Jiaotong University, 710049, Xi\u2019an, Shaanxi, P.R.China"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Zhang", 
        "givenName": "Yulong", 
        "id": "sg:person.011151534565.45", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011151534565.45"
        ], 
        "type": "Person"
      }
    ], 
    "citation": [
      {
        "id": "sg:pub.10.1007/s12044-009-0046-8", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1031722255", 
          "https://doi.org/10.1007/s12044-009-0046-8"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "sg:pub.10.1007/978-1-4757-1738-9", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1033576144", 
          "https://doi.org/10.1007/978-1-4757-1738-9"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "sg:pub.10.1007/s10587-010-0056-9", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1038876523", 
          "https://doi.org/10.1007/s10587-010-0056-9"
        ], 
        "type": "CreativeWork"
      }
    ], 
    "datePublished": "2012-12", 
    "datePublishedReg": "2012-12-01", 
    "description": "For 1 \u2a7e c \u2a7e p \u2212 1, let E1,E2, \u2026,Em be fixed numbers of the set {0, 1}, and let a1, a2, \u2026, am (1 \u2a7d ai \u2a7d p, i = 1, 2, \u2026,m) be of opposite parity with E1,E2, \u2026,Em respectively such that a1a2\u2026am \u2261 c (mod p). Let \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\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}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sum\\limits_{c = 1}^{p - 1} {{E^2}} (c,m,p),$$\\end{document} where E(c, m, p) = N(c,m, p)\u2212((p \u2212 1)m\u22121)/(2m\u22121) for the odd prime p and any integers m \u2a7e 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.", 
    "genre": "article", 
    "id": "sg:pub.10.1007/s10587-012-0068-8", 
    "inLanguage": "en", 
    "isAccessibleForFree": true, 
    "isPartOf": [
      {
        "id": "sg:journal.1135981", 
        "issn": [
          "0011-4642", 
          "1572-9141"
        ], 
        "name": "Czechoslovak Mathematical Journal", 
        "publisher": "Institute of Mathematics, Czech Academy of Sciences", 
        "type": "Periodical"
      }, 
      {
        "issueNumber": "4", 
        "type": "PublicationIssue"
      }, 
      {
        "type": "PublicationVolume", 
        "volumeNumber": "62"
      }
    ], 
    "keywords": [
      "mean value", 
      "A2", 
      "parity", 
      "A1", 
      "A1A2", 
      "number", 
      "values", 
      "analytic methods", 
      "interesting asymptotic formula", 
      "problem", 
      "method", 
      "sum", 
      "formula", 
      "kind", 
      "set", 
      "Lehmer problem", 
      "paper", 
      "asymptotic formula", 
      "integers", 
      "opposite parity", 
      "generalized Lehmer problem"
    ], 
    "name": "On a kind of generalized Lehmer problem", 
    "pagination": "1135-1146", 
    "productId": [
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1043218980"
        ]
      }, 
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/s10587-012-0068-8"
        ]
      }
    ], 
    "sameAs": [
      "https://doi.org/10.1007/s10587-012-0068-8", 
      "https://app.dimensions.ai/details/publication/pub.1043218980"
    ], 
    "sdDataset": "articles", 
    "sdDatePublished": "2022-01-01T18:27", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-springernature-scigraph/baseset/20220101/entities/gbq_results/article/article_563.jsonl", 
    "type": "ScholarlyArticle", 
    "url": "https://doi.org/10.1007/s10587-012-0068-8"
  }
]
 

Download the RDF metadata as:  json-ld nt turtle xml License info

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s10587-012-0068-8'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s10587-012-0068-8'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s10587-012-0068-8'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s10587-012-0068-8'


 

This table displays all metadata directly associated to this object as RDF triples.

101 TRIPLES      22 PREDICATES      50 URIs      39 LITERALS      6 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/s10587-012-0068-8 schema:about anzsrc-for:01
2 anzsrc-for:0101
3 schema:author N5f47bb3e42aa4c79a293e864d77b47ed
4 schema:citation sg:pub.10.1007/978-1-4757-1738-9
5 sg:pub.10.1007/s10587-010-0056-9
6 sg:pub.10.1007/s12044-009-0046-8
7 schema:datePublished 2012-12
8 schema:datePublishedReg 2012-12-01
9 schema:description 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.
10 schema:genre article
11 schema:inLanguage en
12 schema:isAccessibleForFree true
13 schema:isPartOf N3a614476dd114f5aaa61243cc30fc9ff
14 Nf660b9d40ba34c51b658b16f0df1e1a2
15 sg:journal.1135981
16 schema:keywords A1
17 A1A2
18 A2
19 Lehmer problem
20 analytic methods
21 asymptotic formula
22 formula
23 generalized Lehmer problem
24 integers
25 interesting asymptotic formula
26 kind
27 mean value
28 method
29 number
30 opposite parity
31 paper
32 parity
33 problem
34 set
35 sum
36 values
37 schema:name On a kind of generalized Lehmer problem
38 schema:pagination 1135-1146
39 schema:productId N29cad07beb3b4fa2b8f895d153743cac
40 Nc4a1b7f999a1455190f8132bd515e7f3
41 schema:sameAs https://app.dimensions.ai/details/publication/pub.1043218980
42 https://doi.org/10.1007/s10587-012-0068-8
43 schema:sdDatePublished 2022-01-01T18:27
44 schema:sdLicense https://scigraph.springernature.com/explorer/license/
45 schema:sdPublisher N3aa2b88d44564af0a512207538383c01
46 schema:url https://doi.org/10.1007/s10587-012-0068-8
47 sgo:license sg:explorer/license/
48 sgo:sdDataset articles
49 rdf:type schema:ScholarlyArticle
50 N29cad07beb3b4fa2b8f895d153743cac schema:name doi
51 schema:value 10.1007/s10587-012-0068-8
52 rdf:type schema:PropertyValue
53 N3a614476dd114f5aaa61243cc30fc9ff schema:issueNumber 4
54 rdf:type schema:PublicationIssue
55 N3aa2b88d44564af0a512207538383c01 schema:name Springer Nature - SN SciGraph project
56 rdf:type schema:Organization
57 N5f47bb3e42aa4c79a293e864d77b47ed rdf:first sg:person.014071323365.09
58 rdf:rest N6df470567c7f4cb0b494a8bc7d02fb70
59 N6df470567c7f4cb0b494a8bc7d02fb70 rdf:first sg:person.011151534565.45
60 rdf:rest rdf:nil
61 Nc4a1b7f999a1455190f8132bd515e7f3 schema:name dimensions_id
62 schema:value pub.1043218980
63 rdf:type schema:PropertyValue
64 Nf660b9d40ba34c51b658b16f0df1e1a2 schema:volumeNumber 62
65 rdf:type schema:PublicationVolume
66 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
67 schema:name Mathematical Sciences
68 rdf:type schema:DefinedTerm
69 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
70 schema:name Pure Mathematics
71 rdf:type schema:DefinedTerm
72 sg:journal.1135981 schema:issn 0011-4642
73 1572-9141
74 schema:name Czechoslovak Mathematical Journal
75 schema:publisher Institute of Mathematics, Czech Academy of Sciences
76 rdf:type schema:Periodical
77 sg:person.011151534565.45 schema:affiliation grid-institutes:grid.43169.39
78 schema:familyName Zhang
79 schema:givenName Yulong
80 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011151534565.45
81 rdf:type schema:Person
82 sg:person.014071323365.09 schema:affiliation grid-institutes:grid.440588.5
83 schema:familyName Ma
84 schema:givenName Rong
85 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014071323365.09
86 rdf:type schema:Person
87 sg:pub.10.1007/978-1-4757-1738-9 schema:sameAs https://app.dimensions.ai/details/publication/pub.1033576144
88 https://doi.org/10.1007/978-1-4757-1738-9
89 rdf:type schema:CreativeWork
90 sg:pub.10.1007/s10587-010-0056-9 schema:sameAs https://app.dimensions.ai/details/publication/pub.1038876523
91 https://doi.org/10.1007/s10587-010-0056-9
92 rdf:type schema:CreativeWork
93 sg:pub.10.1007/s12044-009-0046-8 schema:sameAs https://app.dimensions.ai/details/publication/pub.1031722255
94 https://doi.org/10.1007/s12044-009-0046-8
95 rdf:type schema:CreativeWork
96 grid-institutes:grid.43169.39 schema:alternateName The School of Electronic and Information Engineering, Xi’an Jiaotong University, 710049, Xi’an, Shaanxi, P.R.China
97 schema:name The School of Electronic and Information Engineering, Xi’an Jiaotong University, 710049, Xi’an, Shaanxi, P.R.China
98 rdf:type schema:Organization
99 grid-institutes:grid.440588.5 schema:alternateName School of Science, Northwestern Polytechnical University, 710072, Xi’an, Shaanxi, P.R.China
100 schema:name School of Science, Northwestern Polytechnical University, 710072, Xi’an, Shaanxi, P.R.China
101 rdf:type schema:Organization
 




Preview window. Press ESC to close (or click here)


...