Abstract:

In number theory, $L$ -functions can be described axiomatically as in the formulation by Selberg or can be viewed as structures arising from interesting arithmetic objects called automorphic representations. The prototypical example of a $L$-function is the Riemann $\zeta$-function. In an 1859 paper, Bernhard Riemann demonstrated how the study of these $L$-functions could potentially solve the mysteries regarding the distribution of prime numbers and also introduced the "holy grail" of number theory, the Riemann Hypothesis. In this talk, we will delve into the analytical facets of these $L$-functions and explore their role in modern mathematics.