Moments of primes in arithmetic progressions

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My research project is concerned with the distribution of primes in arithmetic progressions, that is primes of the form p=qn+a, with a and q two coprime integers which are determined in advance. The distribution of prime numbers is a central question in number theory, and has many applications. Perhaps the most omnipresent practical application of prime numbers is the RSA cryptography algorithm, which is used every time a credit card transaction is being processed on the internet. See for example the nice Slate article http://www.slate.com/articles/health_and_science/science/2013/06/online_credit_card_security_the_rsa_algorithm_prime_numbers_and_pierre_fermat.html. On a more theoretical note, the understanding of primes in arithmetic progressions constitutes a fundamental step in many other number theoretical problems, such as for example the revolutionary theorems of Zhang, Maynard and Tao which imply that there are infinitely many pairs of primes which differ by at most 246. ***My main goal is to understand several statistics (chiefly moments) of the distribution of primes p=qn+a up to a given limit x. The knowledge of all moments is usually sufficient to uniquely determine a distribution. However, the first two moments, the mean and variance, already give a good understanding of a distribution, and imply certain "almost everywhere" statements. These two moments have been extensively studied in the literature by many well-known mathematicians. For instance, Hooley devoted more than fifteen research papers on questions related to the variance. For the first moment, I plan to extend the results of my thesis to different arithmetical contexts, and to introduce Vaughan's approximation in order to obtain more precise results. The range where this last approximation works best is precisely the range where my thesis results apply, and it is clear that sharp results will follow from combining the techniques I used with this approximation. Vaughan introduced his approximation in a paper where he studied the variance of primes in arithmetic progressions. I plan to revise his analysis and to hopefully sharpen his results, by using an alternative technique. As for higher moments, I plan to apply probabilistic arguments using the explicit formula, as was already done with the variance in my previous work. ***The different techniques I will use are quite complementary in that they apply to very different ranges. Probabilistic arguments on explicit formulas work for very small moduli, zero-statistic arguments with the help of random matrix theory work for intermediate moduli, and divisor-switching techniques work for large moduli. The combination of all these techniques should allow for a better understanding of the distribution of primes in arithmetic progressions.**