Spectral Flow for the Riemann zeros

TL;DR

The paper casts the RH problem into a spectral-flow framework within a physics-inspired LM model in which a unitary S-matrix built from the Euler product yields real, σ-dependent energies $E_n(\sigma)$ that flow toward the Riemann zeros as $\sigma\to 1/2^+$. A central result is a simple derivative relation for $E_n(\sigma)$, $\dfrac{d E_n}{d\sigma} = -\dfrac{\Im\Upsilon(s)}{\Re\Upsilon(s) + \vartheta'(t)}$, with $s=\sigma+iE_n(\sigma)$ and $\Upsilon(s)=\dfrac{\zeta'(s)}{\zeta(s)}$, which leads to a criterion: if $-\Re\Upsilon(s) < \vartheta'(t)$ (and a large-$t$ analogue $-\Re\Upsilon(s) \lesssim \tfrac{1}{2}\log(t/2\pi)$) holds for all $\sigma>\tfrac{1}{2}$ and $t>0$, then all $E_n(\sigma)$ remain real, implying RH; this criterion extends naturally to Dirichlet $L$-functions (Generalized RH) and cusp-form $L$-functions (Grand RH). The authors provide numerical evidence supporting the bound for the zeta case, derive analytic connections under RH, and present a counterexample (Davenport–Heilbronn) where the Euler product is absent and the RH fails. They also compare eigenvalue statistics off the critical line to random-matrix predictions, finding GUE behavior only on the critical line. The work suggests a universal, Euler-product–driven route to RH and its generalizations, offering both analytic and numerical validation and highlighting the essential role of unitarity in the underlying scattering picture.

Abstract

Recently, with Mussardo we defined a quantum mechanical problem of a single particle scattering with impurities wherein the quantized energy levels $E_n (σ)$ are exactly equal to the zeros of the Riemann $ζ(s)$ where $σ= \Re (s)$ in the limit $σ\to 1/2$. The S-matrix is based on the Euler product and is unitary by construction, thus the underlying hamiltonian is hermitian and all eigenvalues must be real. Motivated by the Hilbert-Pólya idea we study the spectral flows for $\{ E_n (σ) \}$. This leads to a simple criterion for the validity of the Riemann Hypothesis. The spectral flow arguments are simple enough that we present analogous results for the Generalized and Grand Riemann Hypotheses. We also illustrate our results for a counter example where the Riemann Hypothesis is violated since there is no underlying unitary S-matrix due to the lack of an Euler product.

Figures (16)

• Figure 1: A generic situation for eigenvalues of non-hermitian matrices: two real eigenvalues coalesce to form a pair of complex conjugates.
• Figure 2: Impurities on a circle labeled $j=1, 2,..., N$. Different colors denote different scattering phases.
• Figure 3: A plot of $E_n (\sigma)$ for $n = 1, 2, \ldots, 5$ as a function of $\sigma$ to the right of the critical line.
• Figure 4: A plot $-\Re (\Upsilon (s))$ for $s=\tfrac{3}{4} + i t$ and $\tfrac{1}{2} \log(t/2 \pi)$ as a function of $t$ for $0< t < 30$. The asymptotic version of our bound \ref{['conjecture2']} is only violated for very low $t \lesssim 10$ which is uninteresting.
• Figure 5: A plot $-\Re (\Upsilon (s))$ for $s=\tfrac{3}{4} + i t$ and $\tfrac{1}{2} \log(t/2 \pi)$ as a function of $t$ for $1000< t < 1005$.

Theorems & Definitions (5)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5