*Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.*

Wednesday, March 21, 2018

2:10 p.m.

BA6183

PhD Candidate: Anup Dixit

Supervisor: Kumar Murty

Thesis title: The Lindelof class of L-functions

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Meromorphic functions, called L-functions, play a vital role in number theory. In 1989, Selberg defined a class of L-functions that serves as an axiomatic model for L-functions arising from geometry and arithmetic. Even though the Selberg class successfully captures many characteristics common to most L-functions, it fails to be closed under addition. This creates obstructions, in particular, not allowing us to interpolate between L-functions. To overcome this limitation, V. K. Murty defined a general class of L-functions based on their growth rather than functional equation and Euler product. This class, which is called the Lindelof class of L-functions, is endowed with the structure of a ring.

In this thesis, we study further properties of this class, specifically, its ring structure and topological structure. We also study the zero distribution and the a-value distribution of elements in this class and prove certain uniqueness results, showing that distinct elements cannot share complex values and L-functions in this class cannot share two distinct values with any other meromorphic function. We also establish the value distribution theory for this class with respect to the universality property, which states that every holomorphic function is approximated infinitely often by vertical shifts of an L-function. In this context, we precisely formulate and give some evidence towards the Linnik-Ibragimov conjecture.

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