Grant: Discovery Grants Program - Individual

Analytic number theory is the branch of number theory that studies the natural numbers {1,2,3,4,...} and the prime numbers {2,3,5,7,.} via analysis and complex analysis. Prime numbers have been studied for thousands of years, by all civilizations. Prime numbers are the building blocks of all natural numbers, since any natural number factors into a product  of primes. For instance, 100=2*2*5*5. Despite their simple definition, their occurrence among the other integers remains mysterious and the object of important conjectures in mathematics.  In the last century, prime numbers have become very important for governments and industry. RSA is a method used in internet transactions for sending secret messages securely. This method works because it is extremely difficult to factor a large integer into its prime factors. Analytic number theory arose after publications of Dirichlet in 1837 and Riemann in 1859. These authors showed that prime numbers can be studied via certain complex functions, called L-functions. A very special L-function is the Riemann zeta function. It may be written as an infinite sum of the form 1+1/2^s+1/3^s+..where s is a complex number. An L-function is an similarly defined isum. Both Dirichlet and Riemann realized that L-functions could be used to study prime numbers. Dirichlet showed that there are infinitely many primes having certain special patterns and Riemann showed that their exist formulae for primes in terms of zeros of the zeta function.***Another key advance is due to H. L. Montgomery. In the 1970's he made the fundamental discovery that the zeros of the Riemann zeta function behave like the eigenvalues of large random matrices. He discovered that the statistics of the zeros of the Riemann zeta function are essentially the same as the statistics of the eigenvalues of a large collection of random matrices. He learned of this link from Freeman Dyson, one of the world's leading physicists. In the 1950's, Dyson and other theoretical physicists had been modelling energy levels in quantum chaos by the eigenvalues of large random matrices. ***A third central idea is the combinatorial/sieve method.  This was invented by Erastosthenes and further developed by Brun, Selberg, Rosser, and Bombieri. This methods uses combinatorics to study prime numbers and recently has become very efficient in detecting gaps between primes.  ****My research lies at the interface of the ideas of Dirichlet/Riemann, Montgomery and the sieve method. I propose to study statistical properties of the zeros of L-functions via explicit formula methods. I wish to show there are simple zeros of L-functions, that there are small gaps between zeros of the zeta function, and that on average the zeros of L-functions do not satisfy linear relations. I wish to investigate the connection between zeros of L-functions and the distribution of primes counting sums and related sums. ***<span style=""line-heig