A Generalization to Bellman and Shapiro's Method on the Sum of Digital Sum Functions

Abstract

Consider the digital sum function in base 2. In 1948, R. Bellman and H.N. Shapiro proved a formula for the sum of digital sum functions in base 2 up to a certain number with an error term. However, there was a mistake in their proof. We are able to correct the mistake and walk in their footsteps while retaining their idea and method to prove the theorem. We also generalized their method to prove the same theorem for general base.