Towards quantized number theory: Spectral operators and an asymmetric criterion for the Riemann hypothesis

Abstract

This research expository article not only contains a survey of earlier work but also contains a main new result, which we first describe. Given c ≥ 0, the spectral operator a = ac can be thought of intuitively as the operator which sends the geometry onto the spectrum of a fractal string of dimension not exceeding c. Rigorously, it turns out to coincide with a suitable quantization of the Riemann zeta function ζ = ∂ (s): a = ζ (∂), where ∂ = ∂c is the infinitesimal shift of the real line acting on the weighted Hilbert space L2(ℝ, e-2ct dt). In this paper, we establish a new asymmetric criterion for the Riemann hypothesis (RH), expressed in terms of the invertibility of the spectral operator for all values of the dimension parameter c ∈ (0, 1/2) (i.e. for all c in the left half of the critical interval (0, 1)). This corresponds (conditionally) to a mathematical (and perhaps also, physical) 'phase transition' occurring in the midfractal case when c =1/2. Both the universality and the non-universality of ζ = ζ (s) in the right (resp., left) critical strip {1/2 < Re(s) < 1} (resp.,{0 < Re(s) < 1/2}) play a key role in this context. These new results are presented here. We also briefly discuss earlier joint work on the complex dimensions of fractal strings, and we survey earlier related work of the author with Maier and with Herichi, respectively, in which were established symmetric criteria for the RH, expressed, respectively, in terms of a family of natural inverse spectral problems for fractal strings of Minkowski dimension D ∈ (0, 1), with D ≠ 1/2, and of the quasi-invertibility of the family of spectral operators ac (with c ∈ (0, 1), c ≠ 1/2).