Zeta functions with operator values

Abstract

The famous Riemann zeta function is defined as an infinite series which is convergent in the half plane where the real part is greater than one. It has an analytic continuation to the entire complex plane except at the point one. This can be achieved with a functional equation established by Riemann. The Riemann zeta function also has a huge family of variations or generalizations such as the Dirichlet eta functions and the polylogarithm functions. The Riemann hypothesis conjectures that all the non-trivial zeroes of zeta function lie on the critical line where the real part is equal to one-half. On top of this, Hilbert–Pólya conjecture states that such zeroes should have correspondence to eigenvalues of a self-adjoint operator. Although there have been such operators found that support this conjecture, many puzzles remain to be solved. It is suggested that by allowing these functions to take operator values in certain entries and acting on certain test functions, one is able to explore more on their properties like the distribution of their zeroes.