Analytical and Numerical Study of Inhomogeneous Epstein and Epstein–Hurwitz Zeta Functions

Abstract

In this chapter we present some of the most advanced results obtained in our study of the different zeta functions. From the mathematical point of view they are, without any doubt, quite far reaching and involved. As will be explained in more detail later, the reason why they are not to be found in the mathematical literature dealing with zeta functions (in particular, with Epstein zeta functions) is explained by the fact that non-homogeneous Epstein zeta functions seem not to be very interesting in number theory—contrary to ordinary Epstein zeta functions, which are of paramount importance. The situation in Physics is just the opposite: ordinary Epstein zeta functions appear as a very limited particular case: massless, zero temperature, no chemical potential, of the usual theories. Some of the formulas that will be obtained in the first section are due to the author and are named sometimes after him. They constitute an original and non-trivial extension of the celebrated Chowla–Selberg series formula. Surprisingly enough, the new formulae, which solve the non-homogeneous case (the most important in physical applications) turns out to be, in the end, as beautiful (mathematically) as the one derived by those famous mathematicians for the homogeneous situation. To see the importance that Chowla and Selberg attributed to their discovery, the reader should just throw a look at their original articles. Needless to say the author is equally happy with the new formulae. For physical applications, the importance of these expressions lies in the quick convergence (of exponential type) of the series appearing there. This fact is illustrated at large, in the present chapter, with specific numerical calculations carried out in full detail.