Zeros of L-functions,  9th June - 20th June, 2025 at IISER Kolkata

L-functions are the central objects in Number Theory and one of their remarkable properties is to connect Arithmetic with Analysis, for example the distribution of prime numbers with that of the zeros of the Riemann zeta function. These functions were introduced by Dirichlet in 1837 in his proof of infinitude of primes in arithmetic progressions. Several open questions on the zeros of L-functions are still out of reach though the study of the conjectures has led to the development of a fascinating new theory in the last two centuries. In particular, the location of the zeros of L-functions and their special values have opened up applications within and outside classical number theory. The aim of this course is to familiarize students and young researchers with some aspects of the multiplicative theory of Dirichlet L-functions. We will assume that the participants are beginners of the subject and therefore spend necessary time on preliminaries. Since this GIAN course is inclusive of the students of various educational levels, the first five hours of the lectures and the first five hours of the tutorials are meant to cover all the necessary basics to bridge the knowledge gap between the less-experienced attendees and the advanced learners and to prepare the students to grasp the essence of the seminal works in the later part of the course. This course can be seen as a precursor of a CIMPA school on Automorphic L-functions to be held at IIT Ropar in 30th June to 11th July 2025.