On $\mathcal{B}^4$-almost periodicity for a class of arithmetical functions

Abstract: In this paper, we establish the $\mathcal{B}<sup>4$-almost periodicity in the sense of Besicovitch for a suitably normalized error term associated with a broad class of arithmetical functions introduced by Chandrasekharan and Narasimhan. This result significantly strengthens $\mathcal{B}<sup>2$-almost periodicity previously investigated in the literature for related error terms. By deriving truncated Voronoï-type formulas, we demonstrate that the normalized error term lies in the Besicovitch space $B<sup>4$. As a consequence, we deduce that the error term admits a limit probability distribution and establish an explicit formula for its mean fourth power moment.