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Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.

4 votes

Non-vanishing of zeta(s), Re(s)=1, without complex analysis?

You might want to look at the paper "Le théorème des nombres premiers et la transformation de Fourier" by Jean-Benoît Bost, available at https://www.cmls.polytechnique.fr/xups/xups02-01.pdf It gives a …
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