Dirichlet L-functions, Erdos-Kac theorems, and applications to number theory

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Number theory is one of the most ancient fields of mathematics, often having unexpected application to the real world (such as cryptography and acoustics) as well as other fields of math. Analytic number theory revolves around the single central theme of using the techniques of analysis (such as calculus and complex numbers) to establish information about numbers (such as prime numbers and special sequences). My research involves three aspects of this central theme: zeros of L-functions, normal distribution theorems, and exponential sums.******The distribution of prime numbers is intimately connected to the complex numbers for which certain functions, called L-functions, take 0 as a value. Very little is known about the relationships between the locations of these zeros; for example, can the average of two zeros ever be a third zero? My research will attempt to show that coincidences of this sort never occur. The better we understand these zeros, the more we will be able to determine which sets of numbers contain more prime numbers than other sets, when we group the numbers according to their remainder upon division by some fixed integer.******The normal distribution (or bell curve) is a smooth approximation to a wide variety of data sets, and thus has central importance in both statistics and probability. Surprisingly, many data sets coming from number theory are also approximated by bell curves; for example, picking a large number at random and counting its prime factors results in an approximately normal distribution. My research will greatly expand the types of statistics arising from number theory that have been shown to be approximated by normal distributions.******An exponential sum is the result of adding together many complex numbers, each of which sits on the same circle, at positions determined by the set of numbers we are ultimately trying to study. The better we understand these exponential sums (how large they are, whether there is any long-term trend), the more we can say about the set of numbers the exponential sum is constructed from. My research aims to strengthen our existing knowledge about various types of exponential sums, all for the purpose of pushing old research projects in novel directions.**