On the third moment of $L(\tfrac{1}{2}, \chi_d)$ II: the number field case

TL;DR

This paper derives an asymptotic formula for the third moment of quadratic Dirichlet L-functions at the central point, revealing a secondary term and precise error bounds.

Contribution

It extends previous work by establishing a secondary term of size x^{3/4} in the third moment of quadratic Dirichlet L-functions.

Findings

Main term confirmed for the third moment

Existence of a secondary term of size x^{3/4}

Error term bounded by O(x^{2/3+δ}) for any δ > 0

Abstract

We establish a smoothed asymptotic formula for the third moment of quadratic {D}irichlet $L$-functions at the central value. In addition to the main term, which is known, we prove the existence of a secondary term of size $x^{\frac{3}{4}}$. The error term in the asymptotic formula is on the order of $O(x^{\frac{2}{3}+\delta})$ for every $\delta > 0.$