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I am an undergraduate student writing an expository thesis on the complex-analytic proof of the Prime Number Theorem. I understand that applying the Mellin Transform to the partial sum of the van ...

Consider for $X = 1,2, \ldots$ the set $\mathcal{S}_X$ of trigonometric polynomials $f(t) := \sum_{|k| \leq X} c_k e^{2\pi i kt}$ on the circle $\mathbb{T} := \mathbb{R}/\mathbb{Z}$ of degree $\leq X$ ...

Observe that for any Schwartz function $f \in \mathcal{S}(\mathbb{R})$ having $$ f(0) = \widehat{f}(0) = 1 $$ and $$ f, \widehat{f} \geq 0 \quad \textrm{outside of} \quad [-1,1], $$ the following ...

I am trying to understand the Wiener-Ikehara Tauberian theorem which can be a step to understanding the prime number theorem. Let $$ \hat{a}(s) = \int_0^\infty e^{-us}\, da(u) $$ with $a(u)$ some ...