Ramanujan later used that proof when he discovered Ramanujan primes . In 1932 , Paul ErdÅ ` s published a simpler proof using the Chebyshev function Î ( x ) .
Ramanujan ( 1887 -- 1920 ) used properties of the Gamma function to give a simpler proof , from which the concept of Ramanujan primes would later arise , and ErdÅ ` s ( 1913 â `` 1996 ) in 1932 published a simpler proof using the Chebyshev function , defined as :