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While trying to prove $$\sum_{n \leq x} d(n) = x \log x + (2\gamma - 1) x + O(x^{1/3} \log x) \tag{1}$$

using Voronoi's formula, I had to use the estimate $$K_0(x) \ll e^{-x} \tag{2}$$

as $x \to \infty$, where $K_0$ is the Bessel K function for $\nu = 0$. This can be easily shown once one has the integral representation $$K_0(x) = \int_0^{\infty} \mathrm e^{-x \cosh t} \mathrm dt. \tag{3}$$

How can one show $(3)$ from the original definition of $K_0$ ?

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