Quantum Graphs Whose Spectra Mimic the Zeros of the Riemann Zeta Function

One of the most famous problems in mathematics is the Riemann hypothesis: that the nontrivial zeros of the Riemann zeta function lie on a line in the complex plane. One way to prove the hypothesis would be to identify the zeros as eigenvalues of a Hermitian operator, many of whose properties can be derived through the analogy to quantum chaos. Using this, we construct a set of quantum graphs that have the same oscillating part of the density of states as the Riemann zeros, offering an explanation of the overall minus sign. The smooth part is completely different, and hence also the spectrum, but the graphs pick out the low-lying zeros.

Figure 1

Solid blue: oscillating part (2) of the density of zeros of the Riemann zeta function smoothed with a Gaussian of width $\epsilon =0.4$. The sums over $p$ and $m$ are truncated to $\epsilon m\ln p\le 3.7$. Dotted purple: exact zeros of the Riemann zeta function. Dashed orange: negative of smooth part (1).

Figure 2

Solid blue: density of exact eigenvalues of the set of butterfly graphs (14) smoothed with a Gaussian of width $\epsilon =0.4$ and the mean part subtracted. The set of graphs is truncated at $\epsilon m\ln p\le 3.7$. Dotted purple: exact zeros of the Riemann zeta function. Dashed orange: negative of the smooth part (1) of the Riemann zeros density. Inset: A butterfly graph. Two directed bonds $e_1$ and $e_2$ of the same length $L$ meet at a single vertex, characterized by a $2\times 2$ scattering matrix $S$.