25 Zeta and Related FunctionsApplications25.16 Mathematical Applications25.18 Methods of Computation

Analogies exist between the distribution of the zeros of $\zeta \left(s\right)$ on the critical line and of semiclassical quantum eigenvalues. This relates to a suggestion of Hilbert and Pólya that the zeros are eigenvalues of some operator, and the Riemann hypothesis is true if that operator is Hermitian. See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999).

The zeta function arises in the calculation of the partition function of ideal
quantum gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the
critical gas temperature and density for the Bose–Einstein condensation phase
transition in a dilute gas (Lifshitz and Pitaevskiĭ (1980)). Quantum field theory
often encounters formally divergent sums that need to be evaluated by a process
of regularization: for example, the energy of the electromagnetic vacuum in a
confined space (*Casimir–Polder effect*). It has been found possible to
perform such regularizations by equating the divergent sums to zeta functions
and associated functions (Elizalde (1995)).