Hybrid subconvexity and the partition function

TL;DR

This paper establishes a new hybrid subconvexity bound for certain L-functions, leading to an improved upper bound on the error term in the partition function's Hardy-Ramanujan-Rademacher formula.

Contribution

It introduces a novel hybrid subconvexity bound for central L-values, enhancing the understanding of the partition function's error term.

Findings

Derived a new hybrid subconvexity bound for L(1/2,f×(q/·))

Improved the upper bound for the error term in the partition function

Enhanced the analytic understanding of the partition function's asymptotics

Abstract

We give an upper bound for the error term in the Hardy-Ramanujan-Rademacher formula for the partition function. The main input is a new hybrid subconvexity bound for the central value $L(\tfrac 12,f\times (\tfrac{q}{\cdot}))$ in the $q$ and spectral parameter aspects, where $f$ is a Hecke-Maass cusp form for $\Gamma_0(N)$ and $q$ is a fundamental discriminant.