Z function

Abstract: This chapter discusses the principal problem of the theory of boundary value problems of analytic functions namely the Riemann boundary value problem. It reviews the concept of the index of a function, which is of great value as an auxiliary tool. Various assertions are made in the chapter—(1) the index of a function that is continuous on a closed contour and does not vanish anywhere on it, is an integer or zero; the definition of the index immediately implies the statement, and (2) the index of a product of functions is equal to the sum of the indices of the factors. The index of a quotient is equal to the difference of the indices of the dividend and the divisor. The chapter also discusses the Riemann problem for a simply-connected domain, determination of sectionally analytic function in accordance with given jump, the canonical function of the homogeneous problem, the Riemann problem for the semi-plane, and Riemann boundary value problem with shift.

Abstract: The Riemann problem for a general inhomogeneous system of conservation laws is solved in a neighborhood of a state at which one of the nonlinear waves in the problem takes on a zero speed. The inhomogeneity is modeled by a linearly degenerate field. The solution of the Riemann problem determines the nature of wave interactions, and thus the Riemann problem serves as a canonical form for nonlinear systems of conservation laws. Generic conditions on the fluxes are stated and it is proved that under these conditions, the solution of the Riemann problem exists, is unique, and has a fixed structure; this demonstrates that, in the above sense, resonant inhomogeneous systems generically have the same canonical form. The wave curves for these systems are only Lipschitz continuous in a neighborhood of the states where the wave speeds coincide, and so, in contrast to strictly hyperbolic systems, the implicit function theorem cannot be applied directly to obtain existence and uniqueness. Here we show that existence ...

Abstract: Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here a particular number-theoretical function is chosen, the Riemann zeta function, and its influence on the realm of physics is examined and also how physics may be suggestive for the resolution of one of mathematics' most famous unconfirmed conjectures, the Riemann hypothesis. Does physics hold an essential key to the solution for this more than 100-year-old problem? In this work numerous models from different branches of physics are examined, from classical mechanics to statistical physics, where this function plays an integral role. This function is also shown to be related to quantum chaos and how its pole structure encodes when particles can undergo Bose-Einstein condensation at low temperature. Throughout these examinations light is shed on how the Riemann hypothesis can highlight physics. Naturally, the aim is not to be comprehensive, but rather focusing on the major models and aim to give an informed starting point for the interested reader.