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I am interested in studying the analytic properties of zeta functions and L-functions, especially the distribution of zeros of their derivatives and its relation to the corresponding zeta and L-functions themselves. It is long known that the distribution of zeros of the "Riemann zeta function", the most basic form of zeta functions, is very closely related to the distribution of prime numbers. This breakthrough in the study has attracted many mathematicians and physicists to study this function very closely. Nevertheless, many important properties are left unknown and one of the greatest unsolved problem in mathematics, the "Riemann hypothesis", remains a mystery. The Riemann hypothesis claims that all non-trivial zeros of the Riemann zeta function lie on a straight line. This can be restated as: The first derivative of the Riemann zeta function does not have any zeros in the open left-half of the "critical strip”. We have obtained an analogue of this result in the case of "Dirichlet L-functions", the simplest case of L-functions. I am interested to investigate this further to the case of higher-order derivatives. I am also working on a bigger class of functions, called the “Selberg class”, which contains all zeta functions and L-functions which are expected to satisfy the Riemann hypothesis.
Further, not only restricted to distribution of zeros, I am also studying the distribution of values in general.