# cv.complex variables – Real part of tail of logarithm

Given a positive integer $$n$$, consider $$f_n = -min_{|z|=1} Re sum_{i>n} frac{z^i e^{-i/n}}{i}$$. What can be said about the growth of $$f_n$$? How large can it get?

Taking maximum instead of minimum, the answer is obvious as the maximum is attained at $$z=1$$ and this becomes a real analysis problem (which in particular implies $$f_n$$ is bounded, but perhaps $$f_n$$ tends to $$0$$ at some rate).

An approach suggested by a friend is to express $$sum_{i>n} t^i/i$$ in terms of a Cauchy integral related to $$-ln (1-zeta t)$$ and then perhaps apply a saddle point estimate (uniformly in $$|t|=e^{-1/n}$$) — but I wasn’t able to carry this out.