Toroidal families and averages of \(\mathrm{L}\)-functions. I (Q6595578)

The main result in this first paper is the asymptotic computation of the average of the associated product \(L\)-values, in the case of prime moduli. More precisely, let \(a\) and \(b\) be non-zero integers. There exists an absolute and effective constant \(c > 0\) such that, defining \N\[\N\delta=\frac{c}{|a|+|b|}, \N\]\Nit is shown, for any prime number \(q\), the following asymptotic formulas:\N\begin{itemize}\N\item[(1)] If \(a+b \neq 0\), then \N\[\N\frac{1}{q-1}\sum_{\chi\, (\mathrm{mod}\, q)} L\left(\frac{1}{2},\chi^a\right) L\left(\frac{1}{2},\chi^b\right)=\alpha(a,b)+ O(q^{-\delta}),\N\]\Nwhere \N\[\N\alpha(a,b)=\begin{cases} \zeta\left(\frac{|a|+|b|}{2(a,b)}\right) & \mbox{if } ab<0 \\\N1 & \mbox{if } ab>0. \end{cases} \N\]\N\item [(2)] If \(a+b=0\), then \N\[\N\frac{1}{q-1}\sum_{\chi\, (\mathrm{mod}\, q)} L\left(\frac{1}{2},\chi^a\right) L\left(\frac{1}{2},\chi^{-a}\right)=\log q +2C+O(q^{-\delta}), \N\]\Nwhere \(C\) is a real number independent of \(a\).\N\end{itemize}