Study related related to generalizations of Eiler's constant and their applications

Abstract

Several constants play a dominant role in mathematics due to their appearance in a large number of fundamental identities. Perhaps, some of the most famous examples are the constant which is defined as the ratio of a circle’s circumference to its diameter, Euler’s number e which is defined as the unique number whose natural logarithm is equal to one and the golden ratio ' which is defined as the positive real solution of the quadratic equation x2 ���� x ���� 1 = 0. The present thesis is based on the study of another such important example, the Euler’s constant which was introduced in 1734 by the Swiss mathematician Leonhard Euler in his study on harmonic sums. This constant appears in numerous identities and is considered as fundamental as and e. However, unlike and e we have almost no knowledge about the arithmetic nature of . In particular, it is still unknown whether is rational or irrational. Owing to the mysterious nature of and its frequent occurrence in variety of occasions, the study of and its generalizations is an important topic of research in number theory. As a result, there exist several generalizations of in literature, some of which have been fruitful in shedding some light on our knowledge of and related L-functions. The present thesis is mainly based on further exploration of such generalizations and their connection to L-functions. Firstly, we consider a new class of constants “Shifted Euler constants” and study them along the lines of work done on Euler’s constant in arithmetic progression (r; q) by Briggs, Dilcher, Knopfmacher, Lehmer and some other authors. These constants are then used to give a closed form evaluation of certain integrals involving error term for Dirichlet divisor problem with congruence conditions and also to provide a closed form expression for the value of a class of Dirichlet L-series at any real critical point. Another generalization of which we focus on is the Laurent Stieltjes constants for a principal Dirichlet character 0 . In particular, we study a generalization of the “Generalized Euler constants”,

( ) introduced by Diamond and Ford in 2008 and compare the behaviour of 1( 0) for different modulus of 0. Such constants and investigation of in the context of Hurwitz zeta function enables us to provide a short proof for a closed form expression for the first generalized Stieltjes constant

1(r=q) which has been recently given by Blagouchine in 2015. We also introduce a generalization of ( ) and obtain results in the spirit of work done on by Dilcher, Lehmer, Knopfmacher and some other authors. This kind of generalization is used to obtain a closed form expressions for special values of certain class of Dirichlet L-series. Furthermore, the connection of these constants with a new generalization of Digamma function has been established. This leads us to consider a generalization of a class of generalized Gamma functions introduced by Dilcher in 1994. For such a generalization, we provide a functional equation, Weierstrass product, reflection formulas and some other related properties. Further, we show its relation to the coefficients arising in the Laurent series expansion of partial zeta function at the point s = 1. As an application we derive a fast converging series representation of the generalized Stieltjes constants in terms of some well known functions. In the last part, we recall a famous identity of Gauss which gives a closed form expression for the values of the Digamma function (x) at rational arguments x in terms of elementary functions. These values are intimately connected with a folklore conjecture of Erd˝os which asserts non vanishing of an infinite series associated to a certain class of periodic arithmetic functions. Using such connections and related results of Murty and Saradha , we give a different proof for the identity of Gauss using an orthogonality like relation satisfied by these functions. As a by-product we are able to give a new interpretation for n-th Catalan number in terms of these functions.