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In the 1920s Hecke posed the problem of providing the analytic proof of the reciprocity law for the mth power residue symbol, along the same lines as his proof of relative quadratic reciprocity. We show that a naive approach, based on a suggestive way of generalizing Hecke θ(symbol)-functions, leads to a possibly insurmountable obstacle. However, the generalized θ(symbol)-series we introduce do possess very interesting properties in their own right, including connections with a family of Cauchy-Euler differential equations.
Berg, M. C. “On A Generalization of Hecke ϑ-Functions and the Analytic Proof of Higher Reciprocity Laws.” Journal of Number Theory, vol. 44, no. 1, May 1993, pp. 66–83.