Oscillations of Error Terms Associated with Certain Arithmetical Functions

Abstract.

We prove a general theorem on oscillatory properties of error terms associated with real arithmetical functions whose Mellin transform may have singularities with several summands containing arbitrary complex powers and logarithmic polynomials. The theorem generalizes a theorem of J. Kaczorowski and J. Pintz and is motivated by the applications to algebraic integers with factorizations of distinct lengths. An application of the theorem to the study of the Hilbert semigroup modulo 5 is presented.