Zeros of L-functions, and multiplicative and combinatorial number theory

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Number theory is one of the most ancient fields of pure 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: combinatorial number theory, multiplicative groups in modular arithmetic, and zeros of L-functions. Combinatorial number theory is concerned with understanding additive configurations in sets of integers. For example, do large sets of integers automatically contain a long arithmetic progression - a sequence of integers where the gaps between consecutive integers are all the same? If we add all pairs of integers from a set together, we get a new set; how large must this new set be in terms of the old one? My research investigates this second question in a slightly different setting, namely ordered pairs of integers where only the remainders after division by some prime number are retained. The idea of retaining only the remainders after division by some fixed integer is the key idea in modular arithmetic. If we ignore numbers that have a factor in common with that fixed integer, then the other remainders can be multiplied and divided nicely, forming the fixed integer's "multiplicative group". Groups of this type can be described by certain intrinsic quantities, and my research gives statistical information on how these quantities vary among multiplicative groups. One of the great discoveries of the 19th century is that the distribution of prime numbers is intimately connected to the places where certain functions of complex numbers, 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.