The first moment of quadratic Dirichlet \(L\)-functions at central values (Q6592726)

Let \(\chi_n\) denote the quadratic character \((\frac{\cdot}{n})\) for an odd positive integer \(n\) defined by the Jacobi symbol. In a prior work, \textit{P. Gao} and \textit{L. Zhao} [``First moment of central values of quadratic Dirichlet $L$-functions'', Preprint, \url{arXiv:2303.11588}] evaluated the first moment of the family of \(L^{(2)}(s, \chi_n)\) averaged over all odd positive integers \(n\) under the generalized Riemann hypothesis (GRH). They also deduced the following estimate for the smoothed first moment of \(L\)-functions at central values \[\sum_{(n, 2)=1} L^{(2)}(1/2, \chi_n)\omega\left(\frac{n}{X}\right)=XQ(\log X)+O(X^{1/4+\varepsilon}), \] with a linear polynomial \(Q(x)\) whose expression is omitted. In this paper under review, the author unconditionally evaluates the above first moment of a family of quadratic Dirichlet \(L\)-functions \(L^{(2)}(s, \chi_n)\) averaged over all odd positive integers \(n\) and finally gives an explicit expression of \(Q\).