Riemann Zeta Functions

2003. - 119 pages.A Riemann zeta function is a function which is analytic in the complex plane, with the possible exception of a simple pole at one, and which has characteristic Euler product and functional identity. Riemann zeta functions originate in an adelic generalization of the Laplace transformation which is de ned using a theta function. Hilbert spaces, whose elements are entire functions, are obtained on application of the Mellin transformation. Maximal dissipative transformations are constructed in these spaces which have implications for zeros of zeta functions. The zeros of a Riemann zeta function in the critical strip are simple and lie on the critical line. The Euler zeta function and Dirichlet zeta functions are examples of Riemann zeta functions.