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

Area of Science:

• Mathematics
• Quantum Physics
• Number Theory

Background:

• The Riemann hypothesis is a famous unsolved problem in mathematics.
• It posits that the nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane.
• Proving the hypothesis could involve linking zeta function zeros to eigenvalues of a Hermitian operator.

Purpose of the Study:

• To investigate the connection between the Riemann hypothesis and quantum chaos.
• To construct quantum graphs that model properties of the Riemann zeta function zeros.
• To offer a potential explanation for the behavior of these zeros.

Main Methods:

• Utilizing analogies from quantum chaos theory.
• Constructing a specific set of quantum graphs.
• Comparing the density of states of these graphs to the Riemann zeros.

Main Results:

• The constructed quantum graphs exhibit the same oscillating part of the density of states as the Riemann zeros.
• This provides an explanation for the overall minus sign observed in the Riemann zeta function.
• The smooth part of the density of states differs, but the graphs successfully identify low-lying zeros.

Conclusions:

• Quantum graphs offer a novel framework for studying the Riemann hypothesis.
• The analogy to quantum chaos provides a physical interpretation for some properties of the Riemann zeta function zeros.
• This approach successfully captures key features of the low-lying Riemann zeros.