Arithmetic properties of generalised Euler-Briggs constants [HBNI Th100]

Abstract

This thesis is devoted to study various arithmetic properties of the constants, such as transcendence, linear independence over the field of rational numbers as well as over other number fields and the field of algebraic numbers. In the first chapter we briefly discuss the various facets of Euler's constant and introduce its relevant generalisations that we study in this thesis. In the second chapter, we recall various basic definitions and some of the known results from algebraic, analytic and transcendental number theory which are required for our theorems in the upcoming chapters. At times we indicate briefly the proofs of some of these theorems to keep the exposition self-contained to the extent possible. In the third chapter, we discuss the possible transcendental nature of the generalised Euler-Briggs constants. Some of the main ingredients for the theorems in this chapter are coming from the theory of linear forms in logarithm as developed by A. Baker and the theory of multiplicatively independent cyclotomic units due to K. Ramachandra . In the fourth chapter, we study the linear independence of the generalised Euler-Briggs constants over the field of rational numbers as well as over other number fields and the field of algebraic numbers. We also derive a non-trivial lower bound of certain vector spaces generated by these constants. In addition to the ingredients alluded to above, we shall need a theorem of A. Baker, B. J. Birch and E. A. Wirsing. The penultimate chapter deals with the algebraic independence of these generalised Euler-Briggs constants. The results in this sections are conditional, subject to the weak Schanuel conjecture. In the last chapter, we explore the connection between the generalised Euler-Briggs constants and certain infinite series. Inspired by a result of Lehmer, we derive a necessary and sufficient condition for the existence of periodic Dirichlet series at s = 1 twisted by certain principal Dirichlet character. We express this sum as a linear combination of generalised Euler-Briggs constants. We also prove a result about the special values of a shifted periodic Dirichlet series which can be seen as a variant of the Hurwitz zeta function.