TL;DR
This paper explores the distribution of primes through an extension of the pentagonal number theorem, linking prime distribution to partition functions of specific integer sequences, providing new computational insights.
Contribution
It introduces a new sequence related to integer partitions and extends the pentagonal number theorem to analyze prime distribution.
Findings
Prime distribution relates to a specific partition function.
New sequence allows easier computation of prime-related properties.
Extended pentagonal number theorem offers novel insights into prime patterns.
Abstract
The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application of the partition function to a sequence that we introduce and easily compute in this paper.
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Taxonomy
TopicsAdvanced Mathematical Identities · Analytic Number Theory Research · Advanced Combinatorial Mathematics