) s . Moreover, the function ζ ( s, α ) can be analytically continued to the whole complex plane, except for a simple pole at the point s = 1 with residue 1. For α = 1, the Hurwitz zeta-function becomes the Riemannzeta-function ζ ( s ): ζ ( s
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.
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.
with a natural number k, a non-negative integer j and a complex variable θ, where Δk(x) is the error term in the divisor problem of Dirichlet and Piltz. The main purpose of this paper is to apply the “elementary
methods” and the “elementary formulas” to derive convergence properties and explicit representations of this integral with
respect to θ for k = 2.