Prime numbers and L-functions

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One of the most studied objects in mathematics is the Riemann zeta function. In 1859, the famous German mathematician Bernhard Riemann wrote his one article concerning the theory of numbers. In this important tract he proposed that prime numbers may be studied via the Riemann zeta function. In addition, he provided a specific method to study prime numbers via this function. This method involved studying certain numbers which are called the zeros of the Riemann zeta function. Riemann proposed a simple yet deep conjecture concerning the zeros. The Riemann hypothesis is Riemann's conjecture that all the zeros sit on a straight line. This remains one of the most important unsolved problems in mathematics. It has inspired a significant amount of research by some of the best researchers working in diverse fields of mathematics. Its resolution will provide much finer information concerning our current knowledge of prime numbers. A generalization of this conjecture, the Generalized Riemann Hypothesis, is also believed to be true. This conjecture concerns the location of zeros of L-functions. An L-function is a variant of the Riemann zeta function. My research concerns the behaviour of L-functions and their zeros. These functions also encode deep information concerning prime numbers. I am interested in the interplay between prime numbers and L-functions. In addition, a large part of my research concerns multiplicative functions. These functions are intimately related to both L-functions and primes. Questions concerning the location of zeros of an L-function or the size of an L-function can often be interpreted in terms of a multiplicative function. Therefore I shall investigate the finer behaviour of certain multiplicative functions. In summary, I propose to study L-functions by using tools involving prime numbers and multiplicative functions and on the other hand, I will study primes via properties of L-functions and multiplicative functions.