Inequalities involving Higher Degree Polynomial Functions in $\pi(x)$

TL;DR

This paper investigates inequalities involving higher degree polynomial functions of the prime counting function, providing asymptotic and numerical estimates using advanced number theory tools like the Prime Number Theorem and properties of the Riemann Zeta Function.

Contribution

It introduces new inequalities and estimates for polynomial expressions of $\pi(x)$, expanding understanding of prime distribution and related functions.

Findings

Derived asymptotic estimates for polynomial functions of $\pi(x)$

Established inequalities involving $\pi(x)$ and related functions

Discussed generalizations and applications in prime number theory

Abstract

The primary purpose of this article is to study the asymptotic and numerical estimates in detail for higher degree polynomials in $\pi(x)$ having a general expression of the form, \begin{align*} P(\pi(x)) - \frac{e x}{\log x} Q(\pi(x/e)) + R(x) \end{align*} $P$, $Q$ and $R$ are arbitrarily chosen polynomials and $\pi(x)$ denotes the \textit{Prime Counting Function}. The proofs require specific order estimates involving $\pi(x)$ and the \textit{Second Chebyshev Function} $\psi(x)$, as well as the famous \textit{Prime Number Theorem} in addition to certain meromorphic properties of the \textit{Riemann Zeta Function} $\zeta(s)$ and results regarding its non-trivial zeros. A few generalizations of these concepts have also been discussed in detail towards the later stages of the paper, along with citing some important applications.