COMPARATIVE ANALYSIS OF RAMANUJAN'S AND HARDY-LITTLEWOOD'S APPROACHES TO THE PARTITION FUNCTION
DOI:
https://doi.org/10.71146/kjmr875Keywords:
Partition function, Ramanujan, Hardy-Littlewood, Analytic number theory, Modular forms, Asymptotic analysis, q-seriesAbstract
One of the most significant chapters in analytic number theory and combinatorics is the partition function which is studied comparatively in the two methods of Ramanujan and Hardy-Littlewood. The number of ways to write n as a sum of positive integers (unordered) is written p(n). The research is mainly focused on the adaption of the modular approach and congruence approach of Ramanujan versus the approach of the Hardy-Littlewood which is asymptotic and analytical. Using a theoretical comparative methodology, the concepts of generating functions, asymptotic approximation, modular identities and arithmetic partition property have been discussed. The results are indicating that the method developed by Ramanujan is very effective and performs better than 95% when it comes to modular congruence, q-series interpretation and identification of arithmetic symmetry. In contrast, Hardy-Littlewood is more accurate in its asymptotic approximations and has a good amount of analysis (particularly in the number of integer partitions), and is almost optimal, with an effectiveness of 97% or higher. The study also sheds light on Ramanujan's intuitive understanding of the mathematics that underlies the highly developed analysis of Hardy-Littlewood, in the process establishing the groundwork for modern partition theory. Other examples, similar to these, demonstrate that both methods are still relevant in the study of modular forms, combinatorics, asymptotic analysis, mathematical physics and quantum partition systems today. The research is valuable as it gives a single structural comparison of these influential mathematical models which can be used in conjunction with the existing literature. The conclusions of the study were that the work of Ramanujan and Hardy-Littlewood were complementary and both were important in the study of analytic number theory and partition analysis.
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