# A line integral involving $\log \zeta(s)$

Let $$\zeta$$ denote the Riemann zeta function. Using the Cauchy integral theorem, can you evaluate

$$I=\int_{\Re(s)=\frac{1}{2}} \frac{(2s-1)}{s^{2}(1-s)^2}\Bigg[\int \log((s-1) \zeta(s)) \mathrm{d}s\Bigg] \mathrm{d}s?$$

Note that $$I$$ converges since $$\zeta(s)=O(|s|)$$.

I have provided an answer below as an attempt.

• Assuming this isn't somehow related to Riemann Hypothesis (real part 1/2, and $\zeta$ itself are a bit concerning here)... perhaps if I were to look at it? But you need to show effort, context, etc. – Brevan Ellefsen Dec 11 '18 at 10:30
• @BrevanEllefsen, see my attempted answer. – user507152 Dec 11 '18 at 11:28
• What paper? I sincerely hope you don't mean you are submitting an attempted proof at the RH. Such an attempt would almost surely be ignored and never read if submitted out of nowhere, because so many crack proofs have been submitted. If this is what you mean, please run the paper by a professor first. Otherwise, I wish you the best of luck! – Brevan Ellefsen Dec 11 '18 at 14:37
• There are a few journals that would look at such a paper if it's well written. As I said before, your thoughts look fine on a quick glance, but I can't verify their validity without very careful checking, which I lack time to do. If you intend to submit your argument as a proof you will have to make it super precise and clear to show it's valid. I wish you luck. – Brevan Ellefsen Dec 11 '18 at 14:53

• As you said ''trivial'' we can just ''trivially'' define $S$ so that it excludes $t=0$, in the same vein we excluded the segments $1/2 + i\gamma$ to $\beta + i\gamma$. I've edited the answer to include that. – user507152 Dec 12 '18 at 23:36