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Riemann zeta function is perhaps the most important object of study in number theory. Multiple zeta functions are several variable generalisation of the Riemann zeta function. In the arithmetic side, the study of the special values of these functions is intimately connected to the study of the Riemann zeta values. Whereas, the analytic theory of the Riemann zeta function and the multiple zeta functions are somewhat different. For instance Riemann's functional equation can not be generalised to the multiple zeta functions. We will discuss a novel approach of Ramanujan towards the analytic continuation of the Riemann zeta function, which has led us to study the analytic properties of the multiple zeta functions in the same spirit |