Some time ago while doing formal symbolic manipulations for fun (without worrying about convergence or getting into analysis) to see where I would get, I did the following manipulation:
Starting with the following formula (which I think is quite well known and follows from 3.325 in this book by substitution changing the variable of integration to $\sqrt{x}$):
$$\int_{0}^{\infty}x^{-\frac{1}{2}}e^{-ax-\frac{ab}{x}}dx=\sqrt{\frac{\pi}{a}}e^{-2\sqrt{ab}}$$
I let $a=\ln{n}$ and took a sum from $n=2$ to $\infty$ and then interchanged integration and summation on the left and used the definition of the zeta function to get:
$$\int_{0}^{\infty}x^{-\frac{1}{2}}\left[\zeta\left(x+\frac{b}{x}\right)-1\right]dx=\sqrt{\pi}\sum_{n=2}^{\infty}\frac{1}{n^{2\sqrt{b}}\sqrt{\ln{n}}}$$
I then let $s=2\sqrt{b}$ and performed the substitution $x\rightarrow u^2$ in the integral to arrive at the following interesting formula:
$$\sum_{n=2}^{\infty}\frac{1}{n^{s}\sqrt{\ln{n}}}=\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}\zeta\left(x^{2}+\frac{s^2}{4 x^2}\right)-1 \;dx \tag{1}$$
At a different time, I took $\int_{0}^{\infty}g(t)\sum_\limits{n=2}^{\infty}\frac{1}{n^{st}}dt$ for arbitrary $g(t)$ and rearranged the summation and integration and used the definition of the Laplace transform to turn this into $\sum_\limits{n=2}^{\infty}L[g(t)](s\ln{n})$. I then substituted in $g(t)=H(t-a)f(t-a)$ where $H(t)$ is the Heaviside function and $f(t)$ is arbitrary and used some Laplace transform identities and the definition of the zeta function to get:
$$\int_{0}^{\infty}\left(\zeta(st)-1\right)H(t-a)f(t-a)dt=\sum_{n=2}^{\infty}e^{-as\ln{n}}L[f(t)](s\ln{n})$$
$$=\int_{a}^{\infty}\left(\zeta(st)-1\right) f(t-a)dt$$
Then I set $f(t)=\frac{1}{\pi\sqrt{t}}$ and evaluated the Laplace transform to get:
$$\sum_{n=2}^{\infty}\frac{1}{n^{as}\sqrt{s\ln{n}}}=\int_{a}^{\infty}\frac{\zeta(st)-1}{\sqrt{\pi(t-a)}}dt$$
I then made the substitution $t\rightarrow a+\frac{x^2}{s}$ and set $a=1$ to finally get:
$$\sum_{n=2}^{\infty}\frac{1}{n^{s}\sqrt{\ln{n}}}=\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}\zeta(x^{2}+s)-1 \; dx \tag{2}$$
I think that equations $(1)$ and $(2)$ are quite beautiful, but I do not know whether they are true or not because of the formal kind of way I derived them which can give spurious results as I've found. Although the 2 integrals look similar, I am not sure if they are actually directly related (since I cannot think of a substitution that would turn one into the other) or if the apparent similarity is just a coincidence.
When I tested a few special values of $s$ with Wolfram Alpha, e.g. with $s=\pi$ here, here and here I got that the difference between the two integrals was on the order of $10^{-14}$ and the error in each of the two identities was around $10^{-12}$. Not being familiar with the error in Wolfram Alpha infinite calculations, I am not sure whether that implies that my representations are wrong or not, and I know there can be some very accurate near identities. Other very accurate but not exact approximations I have derived have had errors on the order of $10^{-16}$, which makes me inclined to doubt the expressions' accuracy. I have not been able to find either of these purported identities anywhere.
My question: does anyone know whether identities $(1)$ and $(2)$ are actually true or not, and if they are not does anyone know what the error terms are through a more accurate approach?
