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<?xml version="1.0" encoding="UTF-8" ?> | ||
<root> | ||
<result> | ||
<biographic_references/> | ||
<contributors> | ||
<authors> | ||
<aliases/> | ||
<checked>1</checked> | ||
<codes>maynard.james</codes> | ||
<name>Maynard, James</name> | ||
</authors> | ||
<author_references/> | ||
<editors/> | ||
</contributors> | ||
<document_type>journal article</document_type> | ||
<editorial_contributions> | ||
<reviewer> | ||
<author_code>siaulys.jonas</author_code> | ||
<reviewer_id>11807</reviewer_id> | ||
<name>Jonas Šiaulys</name> | ||
<sign>Jonas Šiaulys (Vilnius)</sign> | ||
</reviewer> | ||
<text>The prime \(k\)-tuples and small gaps between prime numbers are considered. Using a refinement of the Goldston-Pintz-Yildirim sieve method [\textit{D. A. Goldston} et al., Ann. Math. (2) 170, No. 2, 819--862 (2009; Zbl 1207.11096)] the author proves, for instance, the following estimates | ||
\[ | ||
\liminf_{n\to\infty}\,(p_{n+m}-p_n)\ll m^3\text{{e}}^{4m}, \quad \liminf_{n\to\infty}\,(p_{n+1}-p_n)\leq 600 | ||
\] | ||
with an absolute constant in sign \(\ll\). Here \(m\) is a natural number, and \(p_{\,l}\) denote the \(l\)-th prime number.</text> | ||
<contribution_type>review</contribution_type> | ||
</editorial_contributions> | ||
<id>6383667</id> | ||
<keywords>prime number</keywords> | ||
<keywords>small gap</keywords> | ||
<keywords>sieve method</keywords> | ||
<keywords>\(k\)-tuples conjecture</keywords> | ||
<keywords>admissible set</keywords> | ||
<keywords>Selberg sieve</keywords> | ||
<keywords>symmetric polynomial</keywords> | ||
<keywords>symmetric matrix</keywords> | ||
<language> | ||
<languages>English</languages> | ||
<addition/> | ||
</language> | ||
<links> | ||
<identifier>10.4007/annals.2015.181.1.7</identifier> | ||
<type>doi</type> | ||
<url/> | ||
</links> | ||
<links> | ||
<identifier>1311.4600</identifier> | ||
<type>arxiv</type> | ||
<url/> | ||
</links> | ||
<msc> | ||
<code>11N05</code> | ||
<scheme>msc2020</scheme> | ||
<text>Distribution of primes</text> | ||
</msc> | ||
<msc> | ||
<code>11N36</code> | ||
<scheme>msc2020</scheme> | ||
<text>Applications of sieve methods</text> | ||
</msc> | ||
<references> | ||
<doi/> | ||
<position>1</position> | ||
<text>P. D. T. A. Elliott and H. Halberstam, ''A conjecture in prime number theory,'' in Symposia Mathematica, Vol. IV, London: Academic Press, 1970, pp. 59-72.</text> | ||
<zbmath> | ||
<author_codes>elliott.peter-d-t-a</author_codes> | ||
<author_codes>halberstam.heini</author_codes> | ||
<document_id>3377327</document_id> | ||
<msc>11N35</msc> | ||
<msc>11N13</msc> | ||
<prefix>Zbl</prefix> | ||
<series_id>0</series_id> | ||
<year>1970</year> | ||
</zbmath> | ||
</references> | ||
<references> | ||
<doi>10.2307/1971450</doi> | ||
<position>2</position> | ||
<text>J. Friedlander and A. Granville, ''Limitations to the equi-distribution of primes. I,'' Ann. of Math., vol. 129, iss. 2, pp. 363-382, 1989.</text> | ||
<zbmath> | ||
<author_codes>friedlander.john-b</author_codes> | ||
<author_codes>granville.andrew-j</author_codes> | ||
<document_id>4097497</document_id> | ||
<msc>11N05</msc> | ||
<msc>11N13</msc> | ||
<msc>11N35</msc> | ||
<prefix>Zbl</prefix> | ||
<series_id>2531</series_id> | ||
<year>1989</year> | ||
</zbmath> | ||
</references> | ||
<references> | ||
<doi>10.1112/plms/pdn046</doi> | ||
<position>3</position> | ||
<text>D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yildirim, ''Small gaps between products of two primes,'' Proc. Lond. Math. Soc., vol. 98, iss. 3, pp. 741-774, 2009.</text> | ||
<zbmath> | ||
<author_codes>graham.sidney-w</author_codes> | ||
<author_codes>yildirim.cem-yalcin</author_codes> | ||
<author_codes>goldston.daniel-alan</author_codes> | ||
<author_codes>pintz.janos</author_codes> | ||
<document_id>5551831</document_id> | ||
<msc>11N25</msc> | ||
<msc>11N36</msc> | ||
<prefix>Zbl</prefix> | ||
<series_id>628</series_id> | ||
<year>2009</year> | ||
</zbmath> | ||
</references> | ||
<references> | ||
<doi>10.7169/facm/1229442618</doi> | ||
<position>4</position> | ||
<text>D. A. Goldston, J. Pintz, and C. Y. Yildirim, ''Primes in tuples. III. On the difference \(p_{n+\nu}-p_n\),'' Funct. Approx. Comment. Math., vol. 35, pp. 79-89, 2006.</text> | ||
<zbmath> | ||
<author_codes>yildirim.cem-yalcin</author_codes> | ||
<author_codes>pintz.janos</author_codes> | ||
<author_codes>goldston.daniel-alan</author_codes> | ||
<document_id>5135166</document_id> | ||
<msc>11N05</msc> | ||
<msc>11N13</msc> | ||
<prefix>Zbl</prefix> | ||
<series_id>423</series_id> | ||
<year>2006</year> | ||
</zbmath> | ||
</references> | ||
<references> | ||
<doi>10.4007/annals.2009.170.819</doi> | ||
<position>5</position> | ||
<text>D. A. Goldston, J. Pintz, and C. Y. Yildirim, ''Primes in tuples. I,'' Ann. of Math., vol. 170, iss. 2, pp. 819-862, 2009.</text> | ||
<zbmath> | ||
<author_codes>yildirim.cem-yalcin</author_codes> | ||
<author_codes>pintz.janos</author_codes> | ||
<author_codes>goldston.daniel-alan</author_codes> | ||
<document_id>5610431</document_id> | ||
<msc>11N05</msc> | ||
<msc>11N36</msc> | ||
<msc>11N13</msc> | ||
<prefix>Zbl</prefix> | ||
<series_id>2531</series_id> | ||
<year>2009</year> | ||
</zbmath> | ||
</references> | ||
<references> | ||
<doi>10.1112/plms/pdm010</doi> | ||
<position>6</position> | ||
<text>D. A. Goldston and C. Y. Yildirim, ''Higher correlations of divisor sums related to primes. III. Small gaps between primes,'' Proc. Lond. Math. Soc., vol. 95, iss. 3, pp. 653-686, 2007.</text> | ||
<zbmath> | ||
<author_codes>yildirim.cem-yalcin</author_codes> | ||
<author_codes>goldston.daniel-alan</author_codes> | ||
<document_id>5170700</document_id> | ||
<msc>11N05</msc> | ||
<msc>11N37</msc> | ||
<prefix>Zbl</prefix> | ||
<series_id>628</series_id> | ||
<year>2007</year> | ||
</zbmath> | ||
</references> | ||
<references> | ||
<doi/> | ||
<position>7</position> | ||
<text>D. H. J. Polymath, New equidistribution estimates of Zhang type, and bounded gaps between primes.</text> | ||
<zbmath> | ||
<author_codes>polymath.d-h-j</author_codes> | ||
<document_id>6587992</document_id> | ||
<msc>11N35</msc> | ||
<msc>11N05</msc> | ||
<prefix>Zbl</prefix> | ||
<series_id>8474</series_id> | ||
<year>2014</year> | ||
</zbmath> | ||
</references> | ||
<references> | ||
<doi/> | ||
<position>8</position> | ||
<text>A. Selberg, Collected Papers. Vol. II, New York: Springer-Verlag, 1991.</text> | ||
<zbmath> | ||
<author_codes>selberg.atle</author_codes> | ||
<document_id>195021</document_id> | ||
<msc>11-03</msc> | ||
<msc>01A75</msc> | ||
<msc>32-03</msc> | ||
<msc>11M06</msc> | ||
<msc>11M41</msc> | ||
<msc>11N35</msc> | ||
<msc>11N36</msc> | ||
<msc>11F72</msc> | ||
<msc>32N05</msc> | ||
<msc>32N15</msc> | ||
<prefix>Zbl</prefix> | ||
<series_id>0</series_id> | ||
<year>1991</year> | ||
</zbmath> | ||
</references> | ||
<references> | ||
<doi>10.4007/annals.2014.179.3.7</doi> | ||
<position>9</position> | ||
<text>Y. Zhang, ''Bounded gaps between primes,'' Ann. of Math., vol. 179, iss. 3, pp. 1121-1174, 2014.</text> | ||
<zbmath> | ||
<author_codes>zhang.yitang.1</author_codes> | ||
<document_id>6302171</document_id> | ||
<msc>11N05</msc> | ||
<msc>11N13</msc> | ||
<msc>11N35</msc> | ||
<msc>11N36</msc> | ||
<msc>11L07</msc> | ||
<prefix>Zbl</prefix> | ||
<series_id>2531</series_id> | ||
<year>2014</year> | ||
</zbmath> | ||
</references> | ||
<source> | ||
<book/> | ||
<pages>383-413</pages> | ||
<series> | ||
<acronym/> | ||
<issn> | ||
<number>0003-486X</number> | ||
<type>print</type> | ||
</issn> | ||
<issn> | ||
<number>1939-8980</number> | ||
<type>electronic</type> | ||
</issn> | ||
<issue>1</issue> | ||
<issue_id>339578</issue_id> | ||
<parallel_title/> | ||
<part/> | ||
<publisher>Princeton University, Mathematics Department, Princeton, NJ</publisher> | ||
<series_id>2531</series_id> | ||
<short_title>Ann. Math. (2)</short_title> | ||
<title>Annals of Mathematics. Second Series</title> | ||
<volume>181</volume> | ||
<year>2015</year> | ||
</series> | ||
<source>Ann. Math. (2) 181, No. 1, 383-413 (2015).</source> | ||
</source> | ||
<states> | ||
<node>s</node> | ||
<node>item with single author</node> | ||
</states> | ||
<states> | ||
<node>r</node> | ||
<node>item has references</node> | ||
</states> | ||
<states> | ||
<node>c</node> | ||
<node>is cited</node> | ||
</states> | ||
<title> | ||
<addition/> | ||
<original/> | ||
<subtitle/> | ||
<title>Small gaps between primes</title> | ||
</title> | ||
<year>2015</year> | ||
<zbmath_url>https://zbmath.org/6383667</zbmath_url> | ||
<data_source> | ||
<biographic_references>ELASTIC</biographic_references> | ||
<contributors>ELASTIC</contributors> | ||
<document_type>ELASTIC</document_type> | ||
<editorial_contributions>ELASTIC</editorial_contributions> | ||
<id>ELASTIC</id> | ||
<keywords>ELASTIC</keywords> | ||
<language>ELASTIC</language> | ||
<links>ELASTIC</links> | ||
<msc>ELASTIC</msc> | ||
<references>ELASTIC</references> | ||
<states>ELASTIC</states> | ||
<title>ELASTIC</title> | ||
<year>ELASTIC</year> | ||
</data_source> | ||
</result> | ||
<status> | ||
<execution>successful request</execution> | ||
<execution_bool>true</execution_bool> | ||
<internal_code>ok</internal_code> | ||
<query_execution_time_in_seconds>0.04320859909057617</query_execution_time_in_seconds> | ||
<status_code>200</status_code> | ||
<time_stamp>2024-01-08 15:25:14.541364</time_stamp> | ||
</status> | ||
</root> |
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
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import os | ||
import unittest | ||
from xml.dom.minidom import parse | ||
from xmldiff import main, formatting | ||
from xmldiff.actions import MoveNode | ||
from zbmath_rest2oai import getAsXml | ||
|
||
|
||
class PlainXmlTest(unittest.TestCase): | ||
def test_similarity(self): | ||
real_string = getAsXml.final_xml2("6383667") | ||
ref_location = os.path.join(os.path.dirname(__file__), './data/plain.xml') | ||
with open(ref_location) as f: | ||
dom = parse(f) | ||
expected_string = dom.toprettyxml() | ||
diff = main.diff_texts(expected_string, real_string, { | ||
'ratio_mode': 'accurate' | ||
}) | ||
essentials = list(filter(lambda e: not isinstance(e, MoveNode), diff)) | ||
self.assertLessEqual(len(essentials), 333) | ||
diff_text = main.diff_texts(expected_string, real_string, { | ||
'ratio_mode': 'accurate' | ||
}, formatter=formatting.XMLFormatter()) | ||
print(diff_text) | ||
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|
||
if __name__ == '__main__': | ||
unittest.main() |