Class group \(L\)-functions
Let \(K = \mathbb{Q}(\sqrt{-D})\) be an imaginary quadratic field with class number \(h\), and let \(\chi\) be any one of the \(h\) characters of the class group of \(K\). Then the class group \(L\)-functions \(L_K (s,\chi)\) are defined as natural generalizations of the Dedekind zeta-function. The authors consider various mean values of these \(L\)-functions on the critical line, but their ultimate goal is estimating a single \(L\)-function in the \(D\)-aspect. A ``trivial'' estimate is \(\ll D^{1/4} \log^2D\) for fixed \(s = 1/2 + it\) (the implied constant may depend on \(s)\); this follows from the functional equation by the convexity principle. If \(\chi\) is a real character, then \(L_K(s,\chi)\) is a product of two Dirichlet \(L\)-functions, for which well known bounds of Burgess are available. However, the case of complex characters is much more problematic. The average of \(L_K (s,\chi)\) over \(\chi\) reduces to Riemann's zeta-function very precisely. A deeper problem is the mean square, and here a saving by \(D^{-1/28+\varepsilon}\) in the error term is obtained. However, this does not imply any nontrivial estimate for a single \(L\)-function. For this purpose, an ``amplifier'' is attached to the terms of the mean-square; it is a factor which emphasizes the function to be estimated. This idea has been applied by the authors previously to character sums and automorphic \(L\)-functions. The result is \(L_K (s,\chi) \ll D^{1/4-\alpha+\varepsilon}\) with \(\alpha = 1/1156\), but this bound is conditional: three alternative sufficient conditions are given in terms of the class number \(h\), the nontriviality of short field character sums, or the existence of sufficiently many small primes with \(({- D \over p}) = 1\). For instance, the last mentioned property fails for at most finitely many \(D\) in intervals of the type \((X, X^2)\). Also, the estimate holds unconditionally if all the prime factors of \(D\) are bounded by \(D^{\alpha^2}\). The proof is complicated and ingenious; the mean values under consideration are reduced to sums involving automorphic functions at Heegner points, and the spectral theory is used to analyze such sums.
scientific article; zbMATH DE number 809306