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(And a related question: Where should an analytic number theorist learn about Bessel functions?) Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...

Let $f:[0,\infty)\to \mathbb{R}$ be a function with $f(x)=1$ for $0\leq x\leq 1$. Write $Mf$ for the Mellin transform of $f$. Let $c>0$, $T>10^6$ be constants. We are interested in minimizing ...