Generalized Ramanujan Primes View Full Text


Ontology type: schema:Chapter      Open Access: True


Chapter Info

DATE

2014

AUTHORS

Nadine Amersi , Olivia Beckwith , Steven J. Miller , Ryan Ronan , Jonathan Sondow

ABSTRACT

In 1845, Bertrand conjectured that for all integers x ≥ 2, there exists at least one prime in (x∕2, x]. This was proved by Chebyshev in 1860 and then generalized by Ramanujan in 1919. He showed that for any n ≥ 1, there is a (smallest) prime R n such that \(\pi (x) -\pi (x/2) \geq n\) for all x ≥ R n . In 2009 Sondow called R n the nth Ramanujan prime and proved the asymptotic behavior R n ∼ p 2n (where p m is the mth prime). He and Laishram proved the bounds p 2n < R n < p 3n , respectively, for n > 1. In the present paper, we generalize the interval of interest by introducing a parameter c ∈ (0, 1) and defining the nth c-Ramanujan prime as the smallest integer R c, n such that for all x ≥ R c, n , there are at least n primes in (cx, x]. Using consequences of strengthened versions of the Prime Number Theorem, we prove that R c, n exists for all n and all c, that \(R_{c,n} \sim p_{ \frac{n} {1-c} }\) as n → ∞, and that the fraction of primes which are c-Ramanujan converges to 1 − c. We then study finer questions related to their distribution among the primes and see that the c-Ramanujan primes display striking behavior, deviating significantly from a probabilistic model based on biased coin flipping. This model is related to the Cramer model, which correctly predicts many properties of primes on large scales but has been shown to fail in some instances on smaller scales. More... »

PAGES

1-13

Book

TITLE

Combinatorial and Additive Number Theory

ISBN

978-1-4939-1600-9
978-1-4939-1601-6

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-1-4939-1601-6_1

DOI

http://dx.doi.org/10.1007/978-1-4939-1601-6_1

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1004068557


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/2008/0101", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Pure Mathematics", 
        "type": "DefinedTerm"
      }, 
      {
        "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"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "alternateName": "University College London", 
          "id": "https://www.grid.ac/institutes/grid.83440.3b", 
          "name": [
            "Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Amersi", 
        "givenName": "Nadine", 
        "id": "sg:person.012731351123.40", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012731351123.40"
        ], 
        "type": "Person"
      }, 
      {
        "affiliation": {
          "alternateName": "Emory University", 
          "id": "https://www.grid.ac/institutes/grid.189967.8", 
          "name": [
            "Department of Mathematics, Emory University, 404 Dowman Drive, Atlanta, GA\u00a030322, USA"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Beckwith", 
        "givenName": "Olivia", 
        "id": "sg:person.013202531077.42", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.013202531077.42"
        ], 
        "type": "Person"
      }, 
      {
        "affiliation": {
          "alternateName": "Williams College", 
          "id": "https://www.grid.ac/institutes/grid.268275.c", 
          "name": [
            "Department of Mathematics and Statistics, Williams College, Williamstown, MA\u00a001267, USA"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Miller", 
        "givenName": "Steven J.", 
        "id": "sg:person.01101372714.01", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.01101372714.01"
        ], 
        "type": "Person"
      }, 
      {
        "affiliation": {
          "alternateName": "Cooper Union", 
          "id": "https://www.grid.ac/institutes/grid.254672.0", 
          "name": [
            "Department of Electrical Engineering, Cooper Union, New York, NY\u00a010003, USA"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Ronan", 
        "givenName": "Ryan", 
        "id": "sg:person.015121672523.36", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015121672523.36"
        ], 
        "type": "Person"
      }, 
      {
        "affiliation": {
          "name": [
            "209 West 97th Street, New York, NY\u00a010025, USA"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Sondow", 
        "givenName": "Jonathan", 
        "id": "sg:person.07751022527.28", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.07751022527.28"
        ], 
        "type": "Person"
      }
    ], 
    "citation": [
      {
        "id": "https://doi.org/10.1142/s1793042110003848", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1063019675"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.2307/2686886", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1070059627"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.4169/193009709x458609", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1072313401"
        ], 
        "type": "CreativeWork"
      }
    ], 
    "datePublished": "2014", 
    "datePublishedReg": "2014-01-01", 
    "description": "In 1845, Bertrand conjectured that for all integers x\u2009\u2265\u20092, there exists at least one prime in (x\u22152,\u2009x]. This was proved by Chebyshev in 1860 and then generalized by Ramanujan in 1919. He showed that for any n\u2009\u2265\u20091, there is a (smallest) prime R n such that \\(\\pi (x) -\\pi (x/2) \\geq n\\) for all x\u2009\u2265\u2009R n . In 2009 Sondow called R n the nth Ramanujan prime and proved the asymptotic behavior R n \u2009\u223c\u2009p 2n (where p m is the mth prime). He and Laishram proved the bounds p 2n \u2009<\u2009R n \u2009<\u2009p 3n , respectively, for n\u2009>\u20091. In the present paper, we generalize the interval of interest by introducing a parameter c\u2009\u2208\u2009(0,\u20091) and defining the nth c-Ramanujan prime as the smallest integer R c,\u2009n such that for all x\u2009\u2265\u2009R c,\u2009n , there are at least n primes in (cx,\u2009x]. Using consequences of strengthened versions of the Prime Number Theorem, we prove that R c,\u2009n exists for all n and all c, that \\(R_{c,n} \\sim p_{ \\frac{n} {1-c} }\\) as n\u2009\u2192\u2009\u221e, and that the fraction of primes which are c-Ramanujan converges to 1 \u2212 c. We then study finer questions related to their distribution among the primes and see that the c-Ramanujan primes display striking behavior, deviating significantly from a probabilistic model based on biased coin flipping. This model is related to the Cramer model, which correctly predicts many properties of primes on large scales but has been shown to fail in some instances on smaller scales.", 
    "editor": [
      {
        "familyName": "Nathanson", 
        "givenName": "Melvyn B.", 
        "type": "Person"
      }
    ], 
    "genre": "chapter", 
    "id": "sg:pub.10.1007/978-1-4939-1601-6_1", 
    "inLanguage": [
      "en"
    ], 
    "isAccessibleForFree": true, 
    "isPartOf": {
      "isbn": [
        "978-1-4939-1600-9", 
        "978-1-4939-1601-6"
      ], 
      "name": "Combinatorial and Additive Number Theory", 
      "type": "Book"
    }, 
    "name": "Generalized Ramanujan Primes", 
    "pagination": "1-13", 
    "productId": [
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/978-1-4939-1601-6_1"
        ]
      }, 
      {
        "name": "readcube_id", 
        "type": "PropertyValue", 
        "value": [
          "1362b8e2733825cf7a9f8a91f9a55847f51b49691951df255726c6f5e6318e29"
        ]
      }, 
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1004068557"
        ]
      }
    ], 
    "publisher": {
      "location": "New York, NY", 
      "name": "Springer New York", 
      "type": "Organisation"
    }, 
    "sameAs": [
      "https://doi.org/10.1007/978-1-4939-1601-6_1", 
      "https://app.dimensions.ai/details/publication/pub.1004068557"
    ], 
    "sdDataset": "chapters", 
    "sdDatePublished": "2019-04-15T17:11", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000001_0000000264/records_8678_00000245.jsonl", 
    "type": "Chapter", 
    "url": "http://link.springer.com/10.1007/978-1-4939-1601-6_1"
  }
]
 

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.10.1007/978-1-4939-1601-6_1'

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

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/978-1-4939-1601-6_1'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/978-1-4939-1601-6_1'

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

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/978-1-4939-1601-6_1'


 

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

113 TRIPLES      23 PREDICATES      30 URIs      20 LITERALS      8 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/978-1-4939-1601-6_1 schema:about anzsrc-for:01
2 anzsrc-for:0101
3 schema:author N70c785d000c9469faf45feb1e66b09cb
4 schema:citation https://doi.org/10.1142/s1793042110003848
5 https://doi.org/10.2307/2686886
6 https://doi.org/10.4169/193009709x458609
7 schema:datePublished 2014
8 schema:datePublishedReg 2014-01-01
9 schema:description In 1845, Bertrand conjectured that for all integers x ≥ 2, there exists at least one prime in (x∕2, x]. This was proved by Chebyshev in 1860 and then generalized by Ramanujan in 1919. He showed that for any n ≥ 1, there is a (smallest) prime R n such that \(\pi (x) -\pi (x/2) \geq n\) for all x ≥ R n . In 2009 Sondow called R n the nth Ramanujan prime and proved the asymptotic behavior R n  ∼ p 2n (where p m is the mth prime). He and Laishram proved the bounds p 2n  < R n  < p 3n , respectively, for n > 1. In the present paper, we generalize the interval of interest by introducing a parameter c ∈ (0, 1) and defining the nth c-Ramanujan prime as the smallest integer R c, n such that for all x ≥ R c, n , there are at least n primes in (cx, x]. Using consequences of strengthened versions of the Prime Number Theorem, we prove that R c, n exists for all n and all c, that \(R_{c,n} \sim p_{ \frac{n} {1-c} }\) as n → ∞, and that the fraction of primes which are c-Ramanujan converges to 1 − c. We then study finer questions related to their distribution among the primes and see that the c-Ramanujan primes display striking behavior, deviating significantly from a probabilistic model based on biased coin flipping. This model is related to the Cramer model, which correctly predicts many properties of primes on large scales but has been shown to fail in some instances on smaller scales.
10 schema:editor Nf021f7c0d9114996a51eb2d60efd3c4b
11 schema:genre chapter
12 schema:inLanguage en
13 schema:isAccessibleForFree true
14 schema:isPartOf N878a6f0282fa4e66b2e9654369c181b3
15 schema:name Generalized Ramanujan Primes
16 schema:pagination 1-13
17 schema:productId N3ba14f1f9b1e495081e157d9c8d69bd0
18 N594c5710a2614cbaae995bea6313f25b
19 Nfe7c7304a2c64cf28b639afd1f956259
20 schema:publisher Ne40fd0c037ab452f991abe4718cdac66
21 schema:sameAs https://app.dimensions.ai/details/publication/pub.1004068557
22 https://doi.org/10.1007/978-1-4939-1601-6_1
23 schema:sdDatePublished 2019-04-15T17:11
24 schema:sdLicense https://scigraph.springernature.com/explorer/license/
25 schema:sdPublisher N228fb1d973a04f88b44470eeb03204bf
26 schema:url http://link.springer.com/10.1007/978-1-4939-1601-6_1
27 sgo:license sg:explorer/license/
28 sgo:sdDataset chapters
29 rdf:type schema:Chapter
30 N228fb1d973a04f88b44470eeb03204bf schema:name Springer Nature - SN SciGraph project
31 rdf:type schema:Organization
32 N2c9ca44d462a43a3910fc02a4804562b rdf:first sg:person.013202531077.42
33 rdf:rest Na32ec9000663400eba5b706f9eb9c360
34 N34b53f0e41e243ffb7511d7eed2c5d3f schema:name 209 West 97th Street, New York, NY 10025, USA
35 rdf:type schema:Organization
36 N3ba14f1f9b1e495081e157d9c8d69bd0 schema:name doi
37 schema:value 10.1007/978-1-4939-1601-6_1
38 rdf:type schema:PropertyValue
39 N4281cac8be594280b21987244d785b53 rdf:first sg:person.07751022527.28
40 rdf:rest rdf:nil
41 N594c5710a2614cbaae995bea6313f25b schema:name dimensions_id
42 schema:value pub.1004068557
43 rdf:type schema:PropertyValue
44 N70c785d000c9469faf45feb1e66b09cb rdf:first sg:person.012731351123.40
45 rdf:rest N2c9ca44d462a43a3910fc02a4804562b
46 N8265d03ad87a4b74b5dc112c8de96fbc schema:familyName Nathanson
47 schema:givenName Melvyn B.
48 rdf:type schema:Person
49 N878a6f0282fa4e66b2e9654369c181b3 schema:isbn 978-1-4939-1600-9
50 978-1-4939-1601-6
51 schema:name Combinatorial and Additive Number Theory
52 rdf:type schema:Book
53 Na32ec9000663400eba5b706f9eb9c360 rdf:first sg:person.01101372714.01
54 rdf:rest Nfd950c427f274042ad93a0801d0e7ca4
55 Ne40fd0c037ab452f991abe4718cdac66 schema:location New York, NY
56 schema:name Springer New York
57 rdf:type schema:Organisation
58 Nf021f7c0d9114996a51eb2d60efd3c4b rdf:first N8265d03ad87a4b74b5dc112c8de96fbc
59 rdf:rest rdf:nil
60 Nfd950c427f274042ad93a0801d0e7ca4 rdf:first sg:person.015121672523.36
61 rdf:rest N4281cac8be594280b21987244d785b53
62 Nfe7c7304a2c64cf28b639afd1f956259 schema:name readcube_id
63 schema:value 1362b8e2733825cf7a9f8a91f9a55847f51b49691951df255726c6f5e6318e29
64 rdf:type schema:PropertyValue
65 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
66 schema:name Mathematical Sciences
67 rdf:type schema:DefinedTerm
68 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
69 schema:name Pure Mathematics
70 rdf:type schema:DefinedTerm
71 sg:person.01101372714.01 schema:affiliation https://www.grid.ac/institutes/grid.268275.c
72 schema:familyName Miller
73 schema:givenName Steven J.
74 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.01101372714.01
75 rdf:type schema:Person
76 sg:person.012731351123.40 schema:affiliation https://www.grid.ac/institutes/grid.83440.3b
77 schema:familyName Amersi
78 schema:givenName Nadine
79 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012731351123.40
80 rdf:type schema:Person
81 sg:person.013202531077.42 schema:affiliation https://www.grid.ac/institutes/grid.189967.8
82 schema:familyName Beckwith
83 schema:givenName Olivia
84 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.013202531077.42
85 rdf:type schema:Person
86 sg:person.015121672523.36 schema:affiliation https://www.grid.ac/institutes/grid.254672.0
87 schema:familyName Ronan
88 schema:givenName Ryan
89 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015121672523.36
90 rdf:type schema:Person
91 sg:person.07751022527.28 schema:affiliation N34b53f0e41e243ffb7511d7eed2c5d3f
92 schema:familyName Sondow
93 schema:givenName Jonathan
94 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.07751022527.28
95 rdf:type schema:Person
96 https://doi.org/10.1142/s1793042110003848 schema:sameAs https://app.dimensions.ai/details/publication/pub.1063019675
97 rdf:type schema:CreativeWork
98 https://doi.org/10.2307/2686886 schema:sameAs https://app.dimensions.ai/details/publication/pub.1070059627
99 rdf:type schema:CreativeWork
100 https://doi.org/10.4169/193009709x458609 schema:sameAs https://app.dimensions.ai/details/publication/pub.1072313401
101 rdf:type schema:CreativeWork
102 https://www.grid.ac/institutes/grid.189967.8 schema:alternateName Emory University
103 schema:name Department of Mathematics, Emory University, 404 Dowman Drive, Atlanta, GA 30322, USA
104 rdf:type schema:Organization
105 https://www.grid.ac/institutes/grid.254672.0 schema:alternateName Cooper Union
106 schema:name Department of Electrical Engineering, Cooper Union, New York, NY 10003, USA
107 rdf:type schema:Organization
108 https://www.grid.ac/institutes/grid.268275.c schema:alternateName Williams College
109 schema:name Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267, USA
110 rdf:type schema:Organization
111 https://www.grid.ac/institutes/grid.83440.3b schema:alternateName University College London
112 schema:name Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK
113 rdf:type schema:Organization
 




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


...