Hamming weight of powers Given positive integers $b$ and $e$, what is known about the space and time complexity of finding the Hamming weight (number of binary 1s) of $b^e$?

If $e\log b$ bits are available, the number can simply be calculated by standard techniques and the 1s counted.  But what techniques are possible when less memory can be used?
This answer expands my comment above.

You can do it with $O(\log e + \log \log b)$ space as follows:

1) First compute $b^e$ in Chinese remainder representation modulo sufficiently many primes.

2) Then use the Chiu-Davida-Litow algorithm to convert the Chinese remainder representation into binary representation. (Informatique Theoretique et Applications, Vol 35(3), pages 259-275, 2001)

3) Finally, just count the number of $1$'s.

This is a composition of a finite number of log-space computable functions, which is itself log-space computable.