Transcription of THE RIEMANN HYPOTHESIS - Purdue University
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THE RIEMANN HYPOTHESIS - Purdue University
THE RIEMANN HYPOTHESISL ouis de Branges* proof of the RIEMANN HYPOTHESIS is to be obtained for the zeta functionsconstructed from a discrete vector space of finite dimension over the skew field of quaternionswith rational numbers as coordinates in hyperbolic analysis on locally compact Abelian groupsobtained by completion. Zeta functions are generated by a discrete group of symplectictransformations. The coefficients of a zeta function are eigenfunctions ofHecke operatorsdefined by the group. In the nonsingular case the RIEMANN HYPOTHESIS is a consequence ofthe maximal accretive property of a Radon transformation defined in Fourier analysis. In thesingular case the RIEMANN HYPOTHESIS is a consequence of the maximal accretive property ofthe restriction of the Radon transformation to a subspace defined by parity.
The gamma function is an analytic function of s in the complex plane with the exception of singularities at the nonpositive integers which satisfies the recurrence relation sΓ(s) = Γ(s+1). ... isometric property under iterated compositions. The elements of L are functions which
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