An operator theoretic approach to the Riemann Hypothesis

University essay from Lunds universitet/Matematik (naturvetenskapliga fakulteten); Lunds universitet/Matematikcentrum

Author: Ramon Arjan Van De Scheur; [2023]

Keywords: Riemann hypothesis; Prime number theorem; Prime number theorem for arithmetic progressions; operator theoretic equivalent; compact operator; prime-counting function; Chebyshev function; Laplace transform; Fourier transform; convolution operator; weak topology; strong topology; compact perturbation; Tauberian theorem; Poisson kernel; holomorphic; self-adjoint; spectrum; trace; Mathematics and Statistics;

Abstract: In 2023 an operator theoretic approach to the Prime Number Theorem was introduced by Olsen. In this thesis this approach is examined and applied to give a new, operator theoretic, proof of a different version of the Prime Number Theorem and of the Prime Number Theorem for arithmetic progressions. This approach is then expanded and rearranged to focus on the approximation error in the Prime Number Theorem and then operator theoretic equivalents of the Riemann Hypothesis are derived. Lastly, further properties of the operators involved in these operator theoretic equivalents are discussed.

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