A lower bound for the length of addition chains

The length l of addition chains for z is shown to be bounded from below by log₂z+ log₂s(z)?2.13, where s(z) denotes the sum of the digits in the binary expansion of z. The proof given here will also hold for addition-subtraction chains if s(z) is replaced by an appropriate substitute. At first the proof is presented in a simplified version yielding the slightly weaker result l?log₂z+log₂s(z)?0 (log log s(z)).