Critical strings and analyticity of the $\zeta$ function analyticity

Abstract

In this paper we study a simple analytic continuation of the Riemann $\zeta$ function, using Bernoulli numbers and an analytic continuation of the $\Gamma$ function in the complex plane. We use our results to study the critical condition in bosonic string theory. The approach is simple and gives the student an alternative point of view of the subject. We also show that the mathematical basis needed to understand the critical condition is based on well known properties of the Dirichlet series and the theory of entire functions, and is within reach of the average graduate student.