Analytic problems for L-functions and elliptic curves

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This proposed research is centered around L-functions and their applications. L-functions play an important role in modern number theory. In recent years, there has been tremendous interest in the applications of L-functions to studying problems of geometric nature and flavor. In this research I would like to first continue my investigations on L-functions and their applications by considering some analytical problems arising from the theory of elliptic curves. The short term goals include studying the distribution of totally split primes in the division fields of an elliptic curve (and more generally an abelian variety) defined over rationals and studying the number of primes p with fixed number of points on the reduction mod p of an elliptic curve. Secondly this research explores the intimate connections between the zeros of L-functions and the behavior of certain arithmetic functions. For example as a short term goal, under certain standard assumptions on the zeros of the Riemann zeta function, we would like to establish a numerical value for the logarithmic density of the set of positive numbers for which the Mertens conjecture is true. Thirdly this research aims at understanding more about the various L-functions of number theory by studying them in a general setting. A sample problem here is studying sharp upper bounds for the power moments of general L-functions. Such bounds will give us significant insights on the behavior of L-functions. The outcome of this research will enrich our knowledge in several important areas of number theory and will help to pave the way for further studies in the field.