1 Introduction Let s = σ + it be a complex variable, and α , 0 < α ⩽ 1, be a fixed parameter. The Hurwitz zeta-function ζ ( s,α ) was introduced in [ 6 ], and is defined, for σ > 1, by the Dirichlet series ζ ( s , α ) = ∑ m = 0 ∞ 1 ( m + α
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.