Towards Quantized Number Theory: Spectral Operators and an Asymmetric Criterion for the Riemann Hypothesis

This research expository article contains a survey of earlier work (in \S2--\S4) but also contains a main new result (in \S5), which we first describe. Given $c \geq 0$, the spectral operator $\mathfrak{a} = \mathfrak{a}_c$ 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 $\zeta = \zeta(s)$: $\mathfrak{a} = \zeta (\partial)$, where $\partial = \partial_c$ is the infinitesimal shift of the real line acting on the weighted Hilbert space $L^2 (\mathbb{R}, e^{-2ct} dt)$. In this paper, we establish a new asymmetric criterion for the Riemann hypothesis, expressed in terms of the invertibility of the spectral operator for all values of the dimension parameter $c \in (0, 1/2)$ (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 $\zeta = \zeta (s)$ in the right (resp., left) critical strip $\{1/2 < \text{Re}(s) < 1 \}$ (resp., $\{0 < \text{Re}(s) < 1/2 \}$) play a key role in this context. These new results are presented in \S5. In \S2, we briefly discuss earlier joint work on the complex dimensions of fractal strings, while in \S3 and \S4, we survey earlier related work of the author with H. Maier and with H. Herichi, respectively, in which were established symmetric criteria for the Riemann hypothesis, expressed respectively in terms of a family of natural inverse spectral problems for fractal strings of Minkowski dimension $D \in (0,1),$ with $D \neq 1/2$, and of the quasi-invertibility of the family of spectral operators $\mathfrak{a}_c$ (with $c \in (0,1), c \neq 1/2$).