Generalizing the Sum of Digits Function

Abstract

The number theoretic function $G_{q,\alpha } ( n ) = \sum_{k\geqq 1} \sum _{j = 0}^{q - 1} \lfloor n/ q^k + j\alpha \rfloor $ has appeared in the literature for some special values of $\alpha $. Some properties of this function are investigated. Since $G_{q,0} ( n )$ is closely related to the sum of digits of the q-ary representation of n, a generalized “sum of digits” function can be defined via $G_{q,\alpha } $. For $q = 2$ and $\alpha = 2^{ - s} $ the summing function of this “sum of digits” function is analyzed using a technique of Delange.