Bombieri, E. (ed.) et al., Proceedings of the Amalfi conference on analytic number theory, held at Maiori, Amalfi, Italy, from 25 to 29 September, 1989. Salerno: Universitá di Salerno, 367-385 (1992).
Let be a Dirichlet series with for any . Assume that there is an analytic continuation to an entire function, except possibly for a pole at , and suppose there is a functional equation of the usual type. Suppose further that also has a Dirichlet series with supported on the prime powers, and satisfying for some . Various conjectures on such functions are presented, which can be viewed as a very low-brow alternative to the Langlands philosophy.
For example it is conjectured that if and cannot be factorized into other functions of the same type then
where or 0 depending on whether or not.
Subject to certain hypotheses on the zeros of , the value distribution of for fixed near is found, which permits an investigation of the “-points” of (i.e. the zeros of ). Finally similar questions for linear combinations are considered.