Bibliographic Information: Gomez, K. (2024). MacMahonesque Partition Functions Detect Sets Related to Primes. arXiv:2409.14253v2 [math.NT].
Research Objective: This paper investigates the capability of MacMahon partition functions and their generalizations, MacMahonesque functions, to identify sets related to prime numbers beyond simply detecting primes themselves.
Methodology: The author utilizes the framework of quasimodular forms, specifically Eisenstein series, and their properties under the q-differential operator. By constructing specific linear combinations of these forms, the author derives expressions whose Fourier coefficients vanish precisely at the desired sets (cubes of primes or primes in arithmetic progressions).
Key Findings:
Main Conclusions: This work significantly extends the application of MacMahon partition functions in number theory, proving their utility in identifying sets beyond just prime numbers. The author provides concrete examples of these expressions and their corresponding quasimodular forms.
Significance: This research contributes to the growing field of utilizing partition theory to gain insights into classical number theoretic problems. It opens avenues for further exploration of MacMahonesque functions and their potential in detecting other sets of number-theoretic interest.
Limitations and Future Research: The current method is limited to detecting odd powers of primes, specifically cubes, due to constraints in the inequalities used. Further research could explore modifications to overcome this limitation and investigate the detection of higher prime powers. Additionally, exploring the connection between the constructed quasimodular forms and other modular objects could yield deeper insights.
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by Kevin Gomez at arxiv.org 10-10-2024
https://arxiv.org/pdf/2409.14253.pdfDeeper Inquiries