A. Beurling introduced the concept of spectral sets of unbounded functions to study the possibility of the approximation of
those by trigonometric polynomials. We consider spectral sets of unbounded functions in a certain class which contains the
square of the Riemann zeta-function as a typical example.
A. Beurling introduced harmonic functions attached to measurable functions satisfying suitable conditions and defined their
spectral sets. The concept of spectral sets is closely related to approximations by trigonometric polynomials. In this paper
we consider spectral sets of the harmonic functions attached to the Riemann zeta-function and its modification.