Distribution of Primes and Other Topics in Analytic Number Theory

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Goldston 9626903 The distribution of primes in various sequences of integers is of critical importance in Number Theory. The study of this distribution uses both multiplicative and additive number theory. Some of the tools utilized include the circle method, exponential sums, sieves, and the zeta function methods. This project will continue earlier work on obtaining lower bounds for pairs of primes in various sequences. The method uses short divisor sums and has connections with all of the aforementioned tools. The investigator will further develop and generalize this method. He will also examine applications to primes in arithmetic progressions and multiple correlations of primes. A related topic is to obtain asymptotic results when averaging is available. In addition, the investigator will study certain problems involving the power sum method; these problems are well suited to an undergraduate institution. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.