A Study on Arithmetic Nature of q-analogues and p-adics

Abstract

In this thesis, we explore the complex world of mathematics, uncovering a collection of results about the q-analogues of various zeta functions and their interesting properties. Our study is motivated by the remarkable works of Kurokawa and Wakayama in 2003, which introduced a q-variant of the Riemann zeta function, leading to a thorough exploration of these “q” variations. Our exploration begins with a detailed examination of the foundational q-analogue of the Riemann zeta function, represented as ζq(s), defined for q > 1 and ℜ(s) > 1. This function exhibits meromorphic behaviour across the complex plane. Its Laurent series expansion around s = 1 is a main focus of our investigation and it takes the following form: ζq(s) = q −1 log q . 1 s −1 +γ0(q)+γ1(q)(s−1)+γ2(q)(s−1)2 +γ3(q)(s−1)3 +··· . The coefficients γk(q) in this expansion, referred to as q-analogue of the k-th Stieltjes constants, become the building blocks for the subsequent mathematical attempts. The closed-form of these coefficients is derived via intricate formulas, involving Stirling numbers of the first kind, polynomials, and other combinatorial entities, revealing the complexity that underlies their nature. Building upon this foundation, we introduce some results. Few theorems demonstrate the linear independence of the following set of numbers: {1,γ∗ 0(q), γ∗ 0(q2), γ∗ 0(q3), . . . , γ∗ 0(qr)}, where r,q ∈ Z such that r ≥ 1, q > 1, and also involves q-analogue of the Euler’s constant. This leads to a significant improvement on the results by Kurokawa and Wakayama. The transcendence of infinite series involving q-analogue of the first Stieltjes constant, γ1(2), is also established, answering a question posed by Erd˝os in 1948 regarding the arithmetic nature of the infinite series n≥1 σ1(n)/tn, for any integer t > 1. Continuing further, we delve into q-analogues of multiple zeta functions, exploring their behaviour and interrelations. In particular, we calculate a mathematical expression for γ0,0(q), which serves as a “q” version of Euler’s constant with a height of 2. It represents the constant term in the Laurent series expansion of q-version of the double zeta function when centered at s1 = 1 and s2 = 1. Furthermore, we establish results related to linear independence of numbers linked to γ′∗ 0(qi), where 1 ≤ i ≤ r, for any integer r ≥ 1. We also investigate the irrationality of numbers associated with γ0,0(2). Further, as we compare the behaviour of the q-double zeta function when the variables s1 → 0 and s2 → 0 with their classical counterpart, we gain valuable insights into the similarities and distinctions between these functions. Our exploration then advances to introducing several q-variants of the double zeta function, examining their algebraic identities, and uncovering connections among them. These results open new avenues for understanding the intricate relationships between these variants. Taking our research a step further, we turn our attention towards the multi-variable world, introducing a q-variant of the Mordell-Tornheim r-ple zeta function. Furthermore, we also investigate the coefficients of the Laurent series expansion of the q-analogue of the Hurwitz zeta function, which was introduced by Kurokawa and Wakayama in 2003. In the last part, we present a comprehensive study of p-adic analysis, building upon the seminal work of Chatterjee and Gun as a foundational framework. In 2014, Chatterjee and Gun investigated the transcendental nature of special values of the p-adic digamma function, denoted as ψp(r/pn) + γp, for any integer n > 1. Our objective is to extend and generalize these results concerning the transcendental properties of p-adic digamma values. We commence by revisiting a fundamental theorem proposed by them, assert constraints on algebraic elements within a specific set, and highlight the distinctiveness of certain p-adic digamma values. Our research seeks to expand upon this theorem for distinct prime powers and explore the transcendental nature of the p-adic digamma values, with at most one exception. We define and explore various sets, incorporating different prime numbers and scenarios. These theorems establish the transcendental nature of the elements within these sets, with only a limited number of exceptions. Our exploration extends to the realm of composite numbers, specifically focusing on cases, where q ≡ 2 (mod 4). The subsequent theorems shed light on the transcendental properties of p-adic digamma values in this distinct scenario.