Arithmetic Statistics and Analytic Number Theory

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***One of the most fundamental questions in Number Theory is: given an equation (in possibly many variables) with integer coefficients, how many integer solutions does it have? A general answer to this question will have a tremendous impact across the fields of Mathematics, Cryptography, and Computer Science. However, this question turns out to be extremely difficult. For example, we do not know how to answer it even when the equation is a polynomial equation of degree-3 in two variables.******The question becomes somewhat more manageable if instead of considering only one equation, we consider an entire family of equations, and ask about the average number of solutions. The study of solutions across a family of equations is called Arithmetic Statistics.******Equations may also be studied from a different approach. Namely, the solutions to some equations are intimately related to the analytical properties of complex valued functions, called zeta functions or L-functions. The study of such functions, and their relation to these equations is part of what is called Analytic Number Theory.******Having worked in both the fields of Arithmetic Statistics and Analytic Number Theory, my proposal is to combine the tools of these two subjects to tackle fundamental problems in both fields. This will have a broad impact across Number Theory.