Quantum graphs whose spectra mimic the zeros of the Riemann zeta function

Abstract

One of the most famous problems in mathematics is the Riemann hypothesis: that the non-trivial 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.

8 pages, 8 pdf figures

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Cited by in corpus (5)

• Green's function approach for quantum graphs: an overview
• The Riemann zeros as energy levels of a Dirac fermion in a potential built from the prime numbers in Rindler spacetime
• The Riemann zeros as spectrum and the Riemann hypothesis
• Identifying the Riemann zeros by periodically driving a single qubit
• Identifying primes from entanglement dynamics