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An exact formula \[ B(n) = \frac{n}{2}(\lfloor \lg n \rfloor + 1) - \sum_{k=0} ^{\lfloor \lg n \rfloor} 2^k Zigzag(\frac{n}{2^{k+1}}), \] where \[Zigzag (x) = \min (x - \lfloor x \rfloor, \lceil x \rceil - x), \] for theminimal number $ B(n) $ of comparisons of keys performed by $ {\tt MergeSort} $on an $ n $-element array is derived and analyzed. The said formula is lesscomplex than any other known formula for the same and can be evaluated in $O(\log ^{c}) $ time, where $ c $ is a constant. It is shown that there is noclosed-form formula for the above.  Since the recurrence relation for the minimal number of comparisons of keysfor $ {\tt MergeSort} $ is identical with a recurrence relation for the numberof 1s in binary expansions of all integers between $ 0 $ and $ n $(exclusively), the above results extend to the sum of binary digits problem.

Best-case Analysis of MergeSort with an Application to the Sum of Digits Problem, A manuscript (MS) v2