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

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 $\delta >0.$

15 pages