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I guess for the modified Bessel funcion $K_0(z)$, $$\sum_{n=1}^\infty K_0(s\, n) \sim \frac{-2\log 2 - \log \pi + \gamma}{2} + \frac{\log s}{2} + \frac{\pi}{2\, s}, \quad s\to 0^+,$$ if taking $$\...

By the title I mean [reference: ``Spectral methods of Automorphic forms" by Iwaniec (B.41)-(B.43)] for $f\in C^\infty_c(\mathbb{R^+})$, one has $$f(x)=\pi^{-2}\int_{-\infty}^\infty K_{it}(x)F_f(t)t\...

I have asked similar questions on Math Stack Exchange, but not been able to receive many helpful responses. Therefore, I am posting this problem here, and any input would be extremely valuable. I ...

I would like to preface this question by saying that I have asked a series of questions on this topic on Math Stack Exchange, but have almost never received any fruitful responses, with the exception ...

I recently met some problems related to the modified Bessel I funtions. Let $I(\nu,x):=I_\nu(x)$, and $I'_\nu(\nu,x):=\dfrac{\partial}{\partial \nu}I(\nu,x)$. Using maple, it seems that $Re(\dfrac{I'...