Inequalities involving Higher Degree Polynomial Functions in $π(x)$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911992582766592 |
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| author | De, Subham |
| author_facet | De, Subham |
| contents | The primary purpose of this article is to study the asymptotic and numerical estimates in detail for higher degree polynomials in $π(x)$ having a general expression of the form, \begin{align*} P(π(x)) - \frac{e x}{\log x} Q(π(x/e)) + R(x) \end{align*} $P$, $Q$ and $R$ are arbitrarily chosen polynomials and $π(x)$ denotes the \textit{Prime Counting Function}. The proofs require specific order estimates involving $π(x)$ and the \textit{Second Chebyshev Function} $ψ(x)$, as well as the famous \textit{Prime Number Theorem} in addition to certain meromorphic properties of the \textit{Riemann Zeta Function} $ζ(s)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_18983 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inequalities involving Higher Degree Polynomial Functions in $π(x)$ De, Subham General Mathematics Primary 11A41, 11A25, 11N05, 11N37, 11N56, Secondary 11M06, 11M26 The primary purpose of this article is to study the asymptotic and numerical estimates in detail for higher degree polynomials in $π(x)$ having a general expression of the form, \begin{align*} P(π(x)) - \frac{e x}{\log x} Q(π(x/e)) + R(x) \end{align*} $P$, $Q$ and $R$ are arbitrarily chosen polynomials and $π(x)$ denotes the \textit{Prime Counting Function}. The proofs require specific order estimates involving $π(x)$ and the \textit{Second Chebyshev Function} $ψ(x)$, as well as the famous \textit{Prime Number Theorem} in addition to certain meromorphic properties of the \textit{Riemann Zeta Function} $ζ(s)$ 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. |
| title | Inequalities involving Higher Degree Polynomial Functions in $π(x)$ |
| topic | General Mathematics Primary 11A41, 11A25, 11N05, 11N37, 11N56, Secondary 11M06, 11M26 |
| url | https://arxiv.org/abs/2407.18983 |