Research Projects

My research is motivated by classical problems such as the irrationality of values of the Riemann zeta function and its famous generalizations, and it extends into the modern framework of q-analogues and p-adic analysis. Building on my doctoral work titled “A study on arithmetic nature of q-analogues and p-adics”, this proposal aims to investigate the arithmetic properties-particularly irrationality and transcendence-of constants arising in the Laurent expansions of generalized zeta functions and their q-analogues, along with special values of p-adic analogues of other classical functions such as digamma function. Through this proposal, I seek to advance our understanding of the deep algebraic and transcendental structures that underlie these seemingly analytic constructs. In the study of q-series, our results are motivated by the profound works of Kurokawa and Wakayama in 2003. They introduced the q-analogue of the Riemann zeta function as follows: \begin{align} \zeta_q(s) = \sum_{n=1}^{\infty}\frac{q^n}{[n]^s_q}, \end{align} where both q and Re(s) are greater than 1. In the other direction, the research conducted by Chatterjee and Gun in 2014 serves as an inspiration for our study of p-adic theory. Building upon the foundations laid in my doctoral work, my goal is to expand the scope of my investigations within the field of number theory, offering fresh perspectives and advancements. This goal is achieved via intricate formulas, involving Stirling numbers of the first kind, polynomials, and other combinatorial entities, revealing the complexity that underlies their nature. Specifically, I aim to pursue the following directions under the NPDF scheme: Ques 1. What is the arithmetic nature of $\gamma_k(q)$, where $k \geq 2$ and $q \neq 2$? The next questions that can be of potential interest are as follows: Ques 2. Can we derive a comprehensive closed-form expression for $\gamma_{k_1, k_2}(q)$, when both $k_1$ and $k_2$ are non-zero? Ques 3. Could this exploration lead to potential transcendence and irrationality outcomes for these constants? A new direction I intend to initiate is the study of: Ques 4. Arithmetic and analytic properties of derivatives of q-analogues of zeta functions. In addition to this, I am also interested in investigating the coefficients of q-analogues for other variants, such as \begin{itemize} \item The Hurwitz zeta function \item The Mordell-Tornheim zeta function, \end{itemize} as they have remained unexplored. This proposal builds upon significant foundational work in both q-series and p-adic theory and aims to contribute to central questions in number theory through novel techniques. With the opportunity provided by the NPDF scheme, I seek to extend my results and collaborate with experts in related areas to push the boundaries of what is currently known about the arithmetic of special values and their generalizations.