On proving an Inequality of Ramanujan using Explicit Order Estimates for the Mertens Function

2024/07/08 by Subham De, De, Subham
Mathematics · #11A25 #11M26 #11N05 #11N37 #11N56 Secondary 11M06 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #General Mathematics (math.GM) #Mathematical Inequalities and Applications #Primary 11A41

paper · pdf · doi:10.48550/arxiv.2407.12052

openalex publication_date 2024/07/08 · openalex created_date 2024/09/09 · openalex updated_date 2026/07/28

Abstract

This research article provides an unconditional proof of an inequality proposed by Srinivasa Ramanujan involving the Prime Counting Function π(x), (π(x))²lt;(ex)/(log x)π((x)/(e)) for every real x≥ exp(547), using specific order estimates for the Mertens Function, M(x). The proof primarily hinges upon investigating the underlying relation between M(x) and the Second Chebyshev Function, ψ(x), in addition to applying the meromorphic properties of the Riemann Zeta Function, ζ(s) with an intention of deriving an improved approximation for π(x).

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