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 Togbe<ref>{{Cite journal|first1=Yang |last1=Shichun |first2=Togbe |last2=TogbeAlain |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
