The Hardy-Ramanujan-Rademacher expansion for the partition function and its extensions
Mousavi, Seyyed Hamed (2017) The Hardy-Ramanujan-Rademacher expansion for the partition function and its extensions. [Thesis]
Partition theory has been studied more extensively during the last century, athough it has been around since Euler. It is not only because its combinatorial or classical analytical aspects, but also because of the opportunities number theorists saw in applications of modular forms in a di erent and deep view. In this thesis, we study exact formulas for the number of various partitions. For each one, we need to prove a modular transformation formula, and use Farey dissection to avoid the essential singularities of the generating functions. After that, we need to control or estimate the resulting integrals which are rooted from Cauchy integral formula. In this way, we rst study an exact formula for the number of ordinary partitions of any given integer. This formula is a famous result by Ramanujan, Hardy, and Rademacher. Also, we studied another well-known result by Hao, which gives an exact formula for the number of partitions into odd parts. This partition can also be considered for the partitions with distinct parts, thanks to Euler's partition identity. The generating function is a modular form which needs Kloosterman's estimates to handle the integrals. Next, we propose a result which is aimed at the colored partitions with parts of the form 10t±a or 2t±1. This is a continuation of recent works to generalize to partitions into parts in certain symmetric residue classes modulo a given integer. Finally, we will explain about possible future plans to nd exact formulas for various other partition functions.
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