A proof of the Riemann hypothesis

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.