## [0807.0090] A proof of the Riemann hypothesis - arXiv

Abstract. A proof of the Riemann hypothesis is obtained for zeta functions constructed in Fourier analysis on locally compact skew–fields. The skew–fields are the algebra of quater-nions whose coordinates are real numbers and the algebra of quaternions whose coordinates are elements of a p–adic field for every prime p. Fourier analysis is also applied in locally compact algebras which are finite Cartesian products of locally compact skew–fields and in quotient spaces defined by a summation originating in the construction of Jacobian theta functions. The Riemann hypothesis is a consequence of the maximal accretive property of a Radon transformation relating Fourier analysis on a locally compact skew–field with Fourier analysis on a maximal commutative subfield. The maximal accretive property of the Radon transformation is preserved in Cartesian products but need not be preserved in quotient spaces. A proof of the Riemann hypothesis is obtained for zeta functions constructed in quo-tient spaces having the maximal accretive property. When the maximal accretive property fails in a quotient space, the domain of the Radon transformation is decomposed by a symme-try into two invariant subspaces in one of which the maximal accretive property is satisfied. The Riemann hypothesis is proved for the zeta function generated by the quotient space. A

"While I will eventually submit my proof for formal publication, due to the circumstances, I felt it necessary to post the work on the Internet immediately." The Riemann hypothesis is a highly complex theory about the nature of prime numbers - those numbers divisible only by 1 and themselves - that has stymied mathematicians since 1859.

## community to be likely to lead to a proof of the Riemann Hypothesis

I sincerely doubt, however, that he has succeeded with an RH proof this time, either.

Editorial comment: de Branges' apology seems to be a mixture of personal history, mathematical history, and, presumably, the underlying results he will use to obtain proof of the Riemann Hypothesis.