Extension of analytic number theory and the theory of regularized harmonic series from Dirichlet series to Bessel series
Zeta functions (Dirichlet series) are connected to theta functions via the Mellin integral transform. This classical interplay of theta and zeta functions was already used by Riemann to continue the zeta function to the plane. In many aspects of analytical number theory only the logarithmic derivatives of the zeta functions occur, giving rise to an additive theory, which also may be applied to functions which are not logarithmic derivatives. One possible extension, which is the concern of the present paper, is the extension to Bessel series instead of Dirichlet series. In a far reaching program the authors aim at extending the analytical number theory from Dirichlet series to Bessel series. In the present paper they generalize Cramér's theorem, which gives an analytic continuation of the theta series attached to the nontrivial zeros of the Riemann zeta-function. They further give ``explicit formulas'' for Bessel series. In the last sections they include examples from spectral theory and differential geometry which indicate the necessity of using Bessel functions instead of Dirichlet series.
scientific article; zbMATH DE number 931993
scientific article