# Asymptotic expansion of the harmonic numbers

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.

• Well, no worries: that result is weaker than the result you know anyway! – Qiaochu Yuan Feb 26 '16 at 20:03
• To restate slightly: if something is $O(1/z)$, then it is also $O(1/z^{1/4})$. That's because $f(z)\in O(1/z)$ means $|f(z)|/(1/z)=z|f(z)|$ is bounded as $z\to\infty$, and since $z^{1/4}\le z$ for $z\ge 1$, we have that $|f(z)|/(1/z^{1/4})=z^{1/4}|f(z)|$ is also bounded. – ForgotALot Feb 26 '16 at 20:08
• Thanks a lot! I supposed the answer wasn't that hard haha – Sebastian Garrido Feb 26 '16 at 22:50