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{{short description|Prime fulfilling an inequality related to the prime-counting function}}
{{Distinguish|Hardy–Ramanujan number}}
 
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The converse of this result is the definition of Ramanujan primes:
 
:The ''n''th Ramanujan prime is the least integer ''R<sub>n</sub>'' for which <math>\pi(x) - \pi(x/2) \ge n,</math> for all ''x'' ≥ ''R<sub>n</sub>''.<ref>{{MathWorld||authorlinkauthor-link=Jonathan Sondow|author=Jonathan Sondow|title=Ramanujan Prime|urlname=RamanujanPrime}}</ref> In other words: Ramanujan primes are the least integers ''R<sub>n</sub>'' for which there are at least ''n'' primes between ''x'' and ''x''/2 for all ''x'' ≥ ''R<sub>n</sub>''.
 
The first five Ramanujan primes are thus 2, 11, 17, 29, and 41.
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which is the optimal form of ''R''<sub>''n''</sub> ≤ ''c·p''<sub>3''n''</sub> since it is an equality for ''n'' = 5.
 
In a different direction, Axler<ref>{{Cite journal|last=Axler|first=Christian|title=On generalized Ramanujan primes|journal=The Ramanujan Journal|volume=39|issue=2016|pages=1–30|arxiv=1401.7179|year=2014|doi=10.1007/s11139-015-9693-9}}</ref> showed that
 
:<math>R_n < p_{\lceil t\cdot n \rceil}</math>
 
is optimal for ''t'' > 48/19, where <math>\lceil\cdot \rceil</math> is the [[Floor and ceiling functions|ceiling function]].
 
A further improvement of the upper bounds was done in late 2015 by Anitha Srinivasan and John W. Nicholson.<ref>{{Cite journal|first1=Anitha |last1=Srinivasan |first2=John |last2=Nicholson |title=An Improved Upper Bound For Ramanujan Primes|journal=Integers |volume=15 |year=2015 |url=http://www.emis.de/journals/INTEGERS/papers/p52/p52.pdf}}</ref> They show that if
 
:<math>\alpha = 1+\frac{3}{\ln n + \ln \ln n -4}</math>
 
then <math>R_n < p_{\lfloor2n\alpha\rfloor}</math> for all <math> n>241</math>, where <math>\lfloor\cdot\rfloor</math> is the [[Floor and ceiling functions|floor function]]. For
large ''n'', the bound is smaller and thus better than <math>p_{\lfloor2nc\rfloor}</math> for any fixed constant <math>c >1</math>.
 
In 2016, Shichun Yang and Alain Togbé<ref>{{Cite journal|first1=Shichun |last1=Yang |first2=Alain |last2=Togbé |title=On the estimates of the upper and lower bounds of Ramanujan primes|journal=The Ramanujan Journal |volume=40 |issue=2|pages=245–255 |year=2016 |doi=10.1007/s11139-015-9706-8}}</ref> establish the estimates of the [[upper and lower bounds]] of Ramanujan primes <math>R_n</math> when ''n'' is big: if <math>n > 10^{300}</math> and <math>R_n= p_s</math>, then
 
:<math>\beta<s<\alpha, </math>
 
where
 
:<math>\alpha=2n \Big(1+\frac{\ln2}{\ln n}-\frac{\ln2 \ln \ln n-\ln ^2 2-\ln2 -0.13}{\ln ^2 n}\Big), </math>
 
:<math>\beta=2n\Big(1+\frac{\ln2}{\ln n}-\frac{\ln2 \ln \ln n-\ln ^2 2-\ln2 +0.11}{\ln ^2 n}\Big). </math>
 
In 2017 Axler<ref>{{Citation |first=Christian |last=Axler |title=On the number of primes up to the nth Ramanujan prime |year=2017 |arxiv=1711.04588|bibcode=2017arXiv171104588A }}</ref> used similar methods as Yang and Togbé, with improved bounds on the nth prime, to derive improved Ramanujan prime bounds: upper bounds for <math>n \ge 5225</math> and lower bounds for <math>n \ge 1245</math>.
 
== Generalized Ramanujan primes ==
 
Given a constant ''c'' between 0 and 1, the ''n''th ''c''-Ramanujan prime is defined as the
smallest integer ''R<sub>c,n</sub>'' with the property that for any integer ''x ≥ R<sub>c,n</sub>'' there are at least ''n'' primes between ''cx''
and ''x'', that is, <math>\pi(x) - \pi(cx) \ge n</math>. In particular, when ''c'' = 1/2, the ''n''th 1/2-Ramanujan prime is equal to the ''n''th Ramanujan prime: ''R''<sub>0.5,''n''</sub> = ''R<sub>n</sub>''.
 
For ''c'' = 1/4 and 3/4, the sequence of ''c''-Ramanujan primes begins
 
:''R''<sub>0.25,''n''</sub> = 2, 3, 5, 13, 17, ... {{OEIS2C|id=A193761}},
 
:''R''<sub>0.75,''n''</sub> = 11, 29, 59, 67, 101, ... {{OEIS2C|id=A193880}}.
 
It is known<ref>{{Citation |first1=N. |last1=Amersi |first2=O. |last2=Beckwith |first3=S.J. |last3=Miller |first4=R. |last4=Ronan |first5=J. |last5=Sondow |title=Generalized Ramanujan primes |year=2011 |arxiv=1108.0475|bibcode=2011arXiv1108.0475A }}</ref> that, for all ''n'' and ''c'', the ''n''th ''c''-Ramanujan prime ''R<sub>c,n</sub>'' exists and is indeed prime. Also, as ''n'' tends to infinity, ''R<sub>c,n</sub>'' is asymptotic to ''p''<sub>''n''/(1&nbsp;&minus;&nbsp;''c'')</sub>
 
:''R''<sub>''c'',''n''</sub> ~ ''p''<sub>''n''/(1&nbsp;&minus;&nbsp;''c'')</sub> (''n'' → ∞)
 
where ''p''<sub>''n''/(1&nbsp;&minus;&nbsp;''c'')</sub> is the <math>\lfloor</math>''n''/(1&nbsp;&minus;&nbsp;''c'')<math>\rfloor </math>th prime and <math>\lfloor \cdot\rfloor</math> is the [[Floor and ceiling functions|floor]] function.
 
==References==
Retrieved from "https://en.wikipedia.org/wiki/Ramanujan_prime"