Mean values of biquadratic zeta functions (Q1770280)

The author proves an asymptotic formula for a weighted sum of central values of zeta functions of biquadratic extensions of a number field \(K\). If \(K=\mathbb Q\) and \(L={\mathbb Q}(\sqrt{d_1},\sqrt{d_2})\) with \(d_1,d_2\) fundamental discriminants, \((d_1,d_2)=1\), the zeta function of the field \(L\) is \[ \zeta_L(s)=\zeta(s)L(s,\chi_{d_1})L(s,\chi_{d_2})L(s,\chi_{d_1d_2}), \] where \(\zeta=\zeta_{\mathbb Q}\) is the Riemann zeta function. When \(d\) is not a fundamental discriminant, let \(\chi_d\) denote the quadratic character associated to the extension \({\mathbb Q}(\sqrt{d})\) of \({\mathbb Q}\). Let \(L_2(s,\chi_d)\) denote the \(L\)-function with the Euler factor at the prime \(2\) removed. Let \(f\) be a smooth, compactly supported test function satisfying \(\int_0^\infty f(x)\,dx=1\). The main result is \[ \sum_{d_1,d_2>0}a(d_1,d_2)L_2(\frac12,\chi_{d_1}) L_2(\frac12,\chi_{d_2})L_2(\frac12,\chi_{d_1d_2})f(\frac{d_1d_2}{X})\sim \frac{\zeta_2(\frac32)\zeta_2(2)^3}{2\cdot 4!}X\log^4X, \] as \(X\to\infty\). The weighting factor \(a(d_1,d_2)\) satisfies \(a(d_1,d_2)=1\) if \(d_1d_2\) is squarefree, and the weights are small, in the sense that for \(d_1d_2\) squarefree, \[ \sum_{n=1}^\infty\frac{1}{n^{2w}}\Bigl( \sum_{m_1m_2=n^2}a(m_1d_1,m_2d_2)\Bigr) \] converges absolutely for \(\text{Re}(w)>\frac12\).