Digital Sums and Divide-and-Conquer Recurrences: Fourier Expansions and Absolute Convergence

Let ν(n) denote the number of 1’s in the binary representation of n. Properties of this function have been extensively studied in the literature due partly to its natural and frequent appearance in many concrete problems in diverse fields; see [16] and [42] the references therein. For more examples, see [1], [2], [5], [7], [8], [12], [20], [34]. The well-known Trollope-Delange formula (see [13], [46]) for the sum function of ν(n) has attracted much attention in the literature since it represents one of the most concrete examples of producing continuous but nowhere differentiable functions in analysis: for n ≥ 1, nS(n) := n ∑