Exact formula for 1-lower run overpartitions

TL;DR

This work derives an exact Rademacher-type formula for the Fourier coefficients $\overline{p_1}(n)$ counting lower $1$-run overpartitions by extending the Hardy–Ramanujan Circle Method to a mixed mock modular setting. It constructs and analyzes the modular transformation laws of the twisted generating function $\overline{G_1}(q)$, bounds generalized Kloosterman sums, and incorporates Mordell integrals to capture nonholomorphic contributions of mock components. The main result Expresses $\overline{p_1}(n)$ as a convergent sum over $k$ and $\nu$ involving Kloosterman multipliers $K_k^{[12]}$, $K_k^{[22]}$ and integrals $\mathcal{I}_{b,k,\nu}(n)$, and shows the $k=2$ term dominates the asymptotics, yielding a compact leading-term formula. Overall, the paper extends Rademacher-type exact formulas to a new class of mixed-mock modular objects, aligning with prior Circle Method developments for related partition-theoretic functions.

Abstract

We are going to show an exact formula for lower $1$-run overpartitions. The generating function is of mixed mock-modular type with an overall weight $0.$ We will apply an extended version of the classical Circle Method. The approach requires bounding modified Kloosterman sums and Mordell integrals.

Theorems & Definitions (11)

• Example
• Theorem 1.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof