Abstract
In particular, Riemann’s impact on mathematics and physics alike is demonstrated using methods originating from the theory of numbers and from quantum electrodynamics, i.e., from the behavior of an electron in a prescribed external electromagnetic field. More specifically, we employ Riemann’s zeta function to regularize the otherwise infinite results of the so-called Heisenberg–Euler Lagrangian. As a spin-off, we also calculate some integrals that are useful in mathematics and physics.
Highlights
Riemann’s impact on mathematics and physics alike is demonstrated using methods originating from the theory of numbers and from quantum electrodynamics, i.e., from the behavior of an electron in a prescribed external electromagnetic field
Together with the symmetrical form of the functional equation, which was proved by Riemann for all complex s: s s
No wonder that Euler became interested in the fundamentals of its value, e.g. is it rational, transcendental, etc.? Remarkably, it took until the late 20th century before a major breakthrough was achieved in 1979 by the 61-year-old French mathematician Roger Apéry, who was able to prove the irrationality of ζ (3)
Summary
Where ψ( x ) is related to one of Jacobi’s θ functions. Notice that there is no change of the right-hand side s under s → (1 − s). π − 2 Γ 2s ζ (s) has simple poles at s = 0 (from Γ) and s = 1 (from ζ). Π − 2 Γ 2s ζ (s) has simple poles at s = 0 (from Γ) and s = 1 (from ζ) To remove these poles, we multiply by 21 s(s − 1). This is the reason why Riemann defines ξ (s) =. Together with the symmetrical form of the functional equation, which was proved by Riemann for all complex s: 1−s π − 2 ζ (s) = Γ π − 2 (1− s ) ζ ( 1 − s ). Notice that the right-hand side is obtained from the left-hand side by replacing s by 1 − s. Before we continue with Equation (3), we make use of two important formulae due to Euler and Legendre. The latter equation is of great importance
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