A digital sum is the number of occurrences of the digit 1, or some other pattern of digits, in the codes (for some type of code) of 0 , 1 ,..., n - 1. A well-known digital sum is the number of occurrences of the digit 1 in the binary representations of the integers 1 , 2 ,..., n - 1. By the Trollope-Delange formula, this sum is given by S ( n ) = ½ n log2 n + n F0 ( log2 n ), where F0 ( u ) is a continuous, periodic, and nowhere differentiable function that has an explicit Fourier expansion involving the Riemann zeta function. The authors give a new proof for the Trollope-Delange formula based on the Mellin-Perron formulas. Based on this approach, they derive similar exact formulas for other digital sums. This well-written research paper is carefully documented, and comparisons to related work are cited in a helpful 32-item bibliography. Although this work has applications in areas of theoretical computer science such as average case analysis of algorithms, it is more properly a mathematics paper that could best be appreciated by someone with a knowledge of analytic number theory.
