Distributions of prime numbers and zeros of L-functions

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One of the central aspects of analytic number theory is studying the distribution of prime numbers in various sequences of arithmetic interest. Examples of such sequences are arithmetic progressions, values of polynomials, primes shifted by a constant, and so on. Meromorphic functions such as the Riemann zeta function and Dirichlet L-functions and the locations of their zeros are of paramount importance to the understanding of the distribution of primes. In my research, I attempt to quantify the "probability" that one finite arithmetic progression contains more primes than another. I propose to further study these probabilities, first put forward by Rubinstein and Sarnak, from both theoretical and computational viewpoints. I am currently developing a theory that makes precise predictions of their sizes as functions of the arithmetic progressions under consideration, and I plan to test this theory against computations of the probabilities. These computations are nontrivial to say the least, involving lists of zeros of Dirichlet L-functions as well as delicate numerical integrations. I also plan to address the analogous questions for higher-dimensional probabilities that involve, for example, three arithmetic progressions simultaneously. Hypothetical configurations of zeros of the Riemann zeta function or Dirichlet L-functions can imply irregularities in the distribution of prime numbers. I propose to follow the ideas of Ford and Konyagin in determining how such configurations could affect the probabilities described above. Moreover, the recent result of Green and Tao on long arithmetic progressions of primes heralds a breakthrough in methods of analytic number theory. I plan to study possible extensions to their method, in particular how information on zeros of Dirichlet L-functions can be included to address prime values of polynomials, for example.