A Rademacher exact type formula for pod$_2(n)$

TL;DR

This paper derives an exact formula for a specific partition counting function involving restrictions on parts, using advanced analytic techniques related to mock modular forms and the circle method.

Contribution

It introduces a novel exact formula for a restricted partition function and extends the circle method to handle associated mock modular forms.

Findings

Derived an explicit formula for the partition function with even largest part and limited odd repetitions.

Connected the generating function to mixed mock modular forms of weight 0.

Applied advanced bounds on exponential sums and Mordell-type integrals in the analysis.

Abstract

In this paper, we calculate an exact formula for the number of partitions of a natural number $n$, where the largest part is even and no odd parts appears more than two times. The generating functions of the number of these partitions is a mixed mock modular form of weight 0. In order to obtain the formula we apply an extended version of the circle method, during which we need to bound Kloosterman sums and similar exponential sums as well as Mordell-type integrals.