Research Projects

The theory of L-functions is one of the evergreen research areas in Number Theory due to its wide-ranging applications. A well-known example is the Riemann zeta function ζ(s), which is deeply connected to the fundamental properties of the prime numbers. The Dedekind zeta function is an important generalization of ζ(s) and is particularly useful in studying algebraic extensions of the rational numbers. Substantial part of our proposal deals with these L-functions and related areas. The other part examines some problems in q-series and Partitions. Special values of L-functions at specific points play pivotal role in number theory. In this direction, proving ζ(1 + it) is nonzero is equivalent to Prime Number Theorem, a classical result that provides an asymptotic formula for the number of primes up to a given real number. Similarly, the Kronecker limit formula, concerns with constant term in Laurent series expansion of the Dedekind zeta function at s = 1. Determining this constant is important due to its applications, for example, it appears in the special value of an L-function associated with group characters at s = 1, which is in turn related to quantities such as residue of zeta function, class numbers, etc. For example, Kronecker evaluated a twisted zeta function at s = 1 and employed his result in this direction. It is intended to first study special values of various analogues of Dedekind zeta function at s = 1. Recently, in a joint work with Choie, we have dealt with a partial zeta function associated with real quadratic fields. Building upon this work, we now seek similar and higher results for other analogues of Dedekind zeta function along with their applications. Moreover, various special functions appear in these results, such as Herglotz-type functions, whose theory was initiated by Zagier. These special functions are particularly important for their potential applications in constructing Maass cusp forms as Zagier showed that they satisfy certain three-term functional equations. The functions that satisfy three-term relation along with certain growth conditions are known as period functions of Maass cusp forms. We plan to explore these directions. Another active area in number theory is study of moments of ζ(s), motivated in part by the fact they capture information about large values of ζ(1/2 + it) which is then related to Lindelof hypothesis. The transformation formulas for Lambert series often play an important role here. One aim is to derive transformations for general Lambert series and apply them to generalize results on moments of ζ(s). The aforementioned problems are challenging and significant, hence making any progress in these directions is highly important. Similarly, the importance of mock theta functions in q-series and modular forms makes their study essential, as their full understanding remains elusive since Ramanujan introduced them in 1920. One part of this proposal aims to contribute to their ongoing investigation.