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I stumbled upon a random integral and my questions about it are:

1) Why does the Modified Bessel function of the second kind show up in the solution of this integral? Is it because the s in the zeta function is a complex number and the Modified Bessel function is used with complex arguments?

2) What are the steps in order to solve this integral? I have no idea where to start. It doesn't seem clear that the integration methods I learned in calc would apply. Any hints would be welcome!

$$ \int_0^1\zeta(s)^{\frac{1}{\log (x)}}dx = 2\sqrt{\log (\zeta(s))}K_1\left(2 \sqrt{\log (\zeta (s))}\right) ~~~\text{for}~~~ {\rm Re}(\log (\zeta (s))) > 0 $$

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The appearance of $\zeta(s)$ in this expression does not contribute to the result, if ${\rm Re}(\log a) > 0$ then

$$ \int_0^1 a^{1/\log x}{\rm d}x = 2\sqrt{\log a} K_1(2\sqrt{\log a}) $$

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  • $\begingroup$ I know, I was wondering how this integral would be solved $\endgroup$ – Ultradark Apr 13 '18 at 22:10

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