A Note on Some Positivity Conditions Related to Zeta and L-functions

The theory of Hilbert spaces of entire functions [1] was developed by Louis de Branges in the late 1950s and early 1960s with the help of his students including James Rovnyak and David Trutt. It is a generalization of the part of Fourier analysis involving Fourier transform and Plancherel formula. The de Branges-Rovnyak theory of square summable power series, which played an important role in leading to de Branges' discovery of a proof of the Bieberbach conjecture, originated from the theory of Hilbert spaces of entire functions. In [2] de Branges proposed an approach to the generalized Riemann hypothesis, that is, the hypothesis that not only the Riemann zeta function (s) but also all the Dirichlet L-functions L(s; ) with primitive have their nontrivial zeros lying on the critical line <s = 1=2 (See Davenport [5]). In [2] de Branges mentioned that his approach to the generalized Riemann hypothesis using Hilbert spaces of entire functions is related to the Lax-Phillips theory of scattering [6]. In Appendix 2 to Section 7, [6] Lax and Phillips explained the di culty of approaching the Riemann hypothesis by using the scattering theory. In this note, we shall indicate the di culty of approaching the Riemann hypothesis by using de Branges' positivity conditions [2] [3] [4]. In fact, we shall give examples showing that de Branges' positivity conditions, which imply the generalized Riemann hypothesis, are not satis ed by de ning functions of reproducing kernel Hilbert spaces associated with the Riemann zeta function (s) and the Dirichlet L-function L(s; 4).