What question did this study set out to answer?

To establish explicit upper and lower bounds for the prime-counting function with an optimized error-term order and examine their implications for classical open problems in number theory.

August 16, 2026Open Access

The Intrinsic Structure of Primes and Explicit Bounds for (x) with Error‑Term Order O (x/ x)

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PKPing Kuang

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Overview

Theoretical analysis demonstrates explicit tight bounds for the prime-counting function, indicating direct implications for Cramér's conjecture and the Riemann hypothesis.

Key Points

To establish explicit upper and lower bounds for the prime-counting function with an optimized error-term order and examine their implications for classical open problems in number theory.
Constructed explicit lower bound L(x) and upper bound U(x) for the prime-counting function using asymptotic series expansions relative to the logarithmic integral.
Developed a restricted-oscillation prime-counting framework designed to evade classical Littlewood oscillation theorems and Omega-lower-bound constraints.
Demonstrated that the bounding interval satisfies an error-term order of U(x) - L(x) = O(sqrt(x) / log x), with U(x) = Li(x) + O(log x).
Derived a maximal prime-gap formula G(x) = (log x)^2 + O(log x) consistent with Cramér's conjecture.
Showed that the validity of these proposed explicit bounds directly proves the Riemann hypothesis.

Cite This Study

Ping Kuang (2026) studied this question.

synapsesocial.com/papers/6a817aa1f2fb91fc834aeabb https://doi.org/10.5281/zenodo.21941162

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August 16, 2026Open Access

The Intrinsic Structure of Primes and Explicit Bounds for (x) with Error‑Term Order O (x/ x)

View Full Paper

Ask AI

Bookmark

Authors

PKPing Kuang

Discussion

Member takes

Overview

Theoretical analysis demonstrates explicit tight bounds for the prime-counting function, indicating direct implications for Cramér's conjecture and the Riemann hypothesis.

Key Points

To establish explicit upper and lower bounds for the prime-counting function with an optimized error-term order and examine their implications for classical open problems in number theory.
Constructed explicit lower bound L(x) and upper bound U(x) for the prime-counting function using asymptotic series expansions relative to the logarithmic integral.
Developed a restricted-oscillation prime-counting framework designed to evade classical Littlewood oscillation theorems and Omega-lower-bound constraints.
Demonstrated that the bounding interval satisfies an error-term order of U(x) - L(x) = O(sqrt(x) / log x), with U(x) = Li(x) + O(log x).
Derived a maximal prime-gap formula G(x) = (log x)^2 + O(log x) consistent with Cramér's conjecture.
Showed that the validity of these proposed explicit bounds directly proves the Riemann hypothesis.

Cite This Study

Ping Kuang (2026) studied this question.

synapsesocial.com/papers/6a817aa1f2fb91fc834aeabb https://doi.org/10.5281/zenodo.21941162

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