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May 27, 2026Mathematica Slovaca

Note on Legendre-sieve identities at pn+12pₙ₊₁²

Phúc Võ Đăng

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Key Points

This research aims to establish identities related to prime counting functions at specific thresholds.
Developed a finite-sieve framework for analyzing prime counting function π at the square threshold.
Recorded two-sided bounds for π regarding squares of specific primes.
Partitioned composites based on their least prime factor to refine observations.
Established rigorous identities providing improved estimates for π(p_{n+1}^{2}-1) and π(p_{n+1}^{2}).
Provided two-sided bounds for the prime counting function via explicit inequalities.
Compared results with Mertens product at sieve level p_n.

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Implication

Finite-sieve framework assesses prime counting functions in composites, suggesting improved bounds.

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Cite This Study

Phúc Võ Đăng (2026) studied this question.

synapsesocial.com/papers/6a192d4afab5b468c441624a https://doi.org/10.1515/ms-2026-0121

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