Research Projects

(Research Plan A: Equivalent criteria for the Riemann hypothesis). The theory of the Riemann zeta function plays a central role in the development of analytic number theory. Riemann conjectured that all the non-trivial zeros of the Riemann zeta function zeta(s) will lie on the critical line Re(s)=1/2. This conjecture is popularly known as the Riemann hypothesis (RH). Over the years, mathematicians gave many useful equivalent criteria for the Riemann hypothesis while trying to prove it. One of the notable equivalent criteria is due to Littlewood. In 1916, Riesz found an equivalent criterion for RH. Around the same time, Hardy and Littlewood established another equivalent criterion for RH while correcting an identity of Ramanujan. Recently, together with my Ph.D. students A. Agarwal and M. Garg, we found a one-variable generalization of the identity of Hardy and Littlewood. Moreover, as an application of the generalized identity, we found equivalent criteria for RH. In particular, our bound generalizes the bound given by Hardy and Littlewood as well as the bound of Riesz. In 2012, A. Dixit derived a Dirichlet character analogue of the Hardy-Littlewood identity. Recently, together with M. Garg, we established a one-variable generalization of the identity of A. Dixit. Moreover, we found equivalent criteria for the generalized Riemann hypothesis for the Dirichlet L-function. In the same paper, we gave a general conjecture for nice L-functions. (Research Plan B: Number field analogue of Ramanujan’s formula for zeta(2m+1)). Euler’s formula for zeta(2m) implies that all even zeta values are transcendental. However, we do not know much about the nature of odd zeta values except zeta(3). Roger Apery proved the irrationality of zeta(3). In 2001, Rivoal, Ball and Rivoal made a breakthrough by proving that there exist infinitely many odd zeta values which are irrational. Around the same time, Zudilin showed that at least one of zeta(5), zeta(7), zeta(9), or zeta (11) is irrational. Ramanujan’s notebooks contain many elegant identities for zeta(s) and one of the identities is a formula for zeta(2m+1) which attracted many mathematicians. From Ramanujan’s identity, one can obtain transformation formula for the Eisenstein series over the full modular group. Recently, together with A. Dixit, we found a new generalization of Ramanujan's formula. Very recently, together with my M.Sc. student Anushree Gupta, we showed that Ramanujan’s formula for zeta(1/2) and zeta(2m+1) can be derived from a more general identity. The Dedekind zeta function is a number field analogue of the Riemann zeta function. A number field analogue of Euler’s formula for zeta(2m) has been independently established by Siegel and Klingen in case of a totally real field. However, we do not know an analogue of Euler’s formula for imaginary number fields.