On Proving Ramanujan's Inequality using a Sharper Bound for the Prime Counting Function π(x)

2024/08/05 by De, Subham
#11A25 #11N05 #11N37 Secondary 11Y99 #FOS: Mathematics #Number Theory (math.NT) #Primary 11A41

paper · doi:10.48550/arxiv.2408.02591

Abstract

This article provides a proof that the Ramanujan's Inequality given by, π(x)² lt; (e x)/(log x) π((x)/(e)) holds unconditionally for every x≥ exp(43.5102147). In case for an alternate proof of the result stated above, we shall exploit certain estimates involving the Chebyshev Theta Function, ϑ(x) in order to derive appropriate bounds for π(x), which'll lead us to a much improved condition for the inequality proposed by Ramanujan to satisfy unconditionally.

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