Hey Math Stack Exchange,
I have an integral that I'd like to try and simplify or get bounds on. It involves the Riemann Zeta function which I'm not super familiar with and so I thought I'd look for help here since you all have been helpful in the past. Thank you for taking a look at it and any help would be greatly appreciated!
$$f(x) = \Big(\frac{1}{2 \pi i}\Big)*\int_{c-i \infty}^{c + i \infty} \frac{x^w}{w}*\sum_{k=2}^{\infty}\frac{\zeta^k(w)}{k} dw$$
where $c=2\gamma -1$ where $\gamma$ is the Euler Mascheroni Constant.
There are a couple of things I'm trying to understand about it:
1) Is there a simplified form or an identity?
2) Is there a lower bound or upper bound behavior to it? (i.e. Big-O or Little-O)?
3) What happens to this as we take the limit of x going to infinity?
Thank you guys again for taking a look at it!