Asymptotic expansion of the harmonic numbers

Question

I was skimming through Atle Selberg's "Elementary Proof of the Prime Number Theorem", and I got stumped at the part where he introduced equation 2.7 $(\sum_{v\leq z} \frac{1}{v} = log z + c_{1} + O(z^{-1/4}))$ (of which he only says that it is well known). I've seen a similar equation and it's proof $(\sum_{v\leq z} \frac{1}{v} = log z + \gamma + O(\frac{1}{z}))$, but I can't find the former anywhere else.