Consider the following weighted digital sum (WDS) variant: write integer $n$ as $n=2^{i_1} + 2^{i_2} + \cdots + 2^{i_k}$ with $i_1 > i_2 > \cdots > i_k \geq 0$ and set $W_M(n) := \sum_{t=1}^k t^M 2^{i_t}$. This type of weighted digital sum arises (when $M=1$) in the analysis of bottom-up mergesort but is not "smooth'' enough to permit a clean analysis. We therefore analyze its average $TW_M(n) := \frac{1}{n}\sum_{j \gt n} W_M(j)$. We show that $TW_M(n)$ has a solution of the form $n G_M(\lg n) + d_M \lg ^M n + \sum\limits_{d=0}^{M-1}(\lg ^d n)G_{M,d}(\lg n)$, where $d_M$ is a constant and $G_M(u), G_{M,d}(u)$'s are periodic functions with period one (given by absolutely convergent Fourier series).
Discrete Mathematics & Theoretical Computer Science