A study of arithmetic nature of certain number theoretic constants and their special values

Abstract

In 1734, Euler introduced the constant popularly known as Euler's constant or Euler-Mascheroni constant as the limiting di erence between harmonic series and natural logarithm:

:= lim x!1 X n x 1 n 􀀀 log x

: After Euler, several other prominent mathematicians have studied this constant in depth and this constant has occurred in various expressions such as the Laurent series expression of the Riemann zeta function around s = 1, lower bounds to a prime gap, product formula for gamma function etc. Because of the mysterious nature of and its constant appearance in various expressions, it makes it a fundamental object to study in number theory. Besides its appearance in various equations, the number has not been proved algebraic or transcendental. In fact it is not known whether

is rational or not. Once G. H. Hardy said to have stated that he would give up his endowed chair at Oxford to anyone who could prove to be irrational. Though one can expect it to be a transcendental number and thanks to a conjecture of Kontsevich and Zagier [38] in which they predicted that is not even a period. Thus, instead of studying it independently many mathematicians have tried to study a family of numbers where is a member. The present thesis is based on the study of arithmetic nature and linear independence of some special type of numbers known as harmonic numbers, logarithms of cyclotomic numbers, special values of the digamma function and logarithms of gamma function at rational arguments. One such family of numbers is the classical digamma function at rational arguments where the digamma function is de ned as (x) := d dx log(􀀀(x)) = 􀀀0(x) 􀀀(x) : It is well known that (x) is a meromorphic function with simple poles at the non-negative integers with residue 􀀀1. Similar to that of the gamma function, the digamma function also satis es some interesting properties such as (1 􀀀 x) = (x) + cot( x) Re ection formula, (x + 1) = (x) + 1 x Recurrence relation. The arithmetic nature of the digamma function at rational arguments has been the talk of the town for a long time which is important to study the special values of L-functions at s = 1. One major contribution in studying the nature of special values of digamma function at rational arguments is due to the following celebrated formula of Gauss [25] in 1813: (a=q) = 􀀀 􀀀 log(2q) 􀀀

2 cot

a q

+ 2 b q􀀀1 X2 c k=1 cos

2a k q

log sin

k q

: (1) However, the mysterious nature of the constant makes its di cult to investigate the nature of the special values of the digamma function at rational arguments. In order to avoid this di culty one can ask the arithmetic nature of the family of numbers (a=q) + instead of (a=q). In 2007, Murty and Saradha (for a proof see [47]) investigated the transcendental nature of the numbers (a=q) + ; where 1 a q 􀀀 1. Along with the nature of (a=q) + values, they also proved some interesting results on the arithmetic nature and linear independence of the digamma values. In fact on the other hand when is a part of the formula in Eq. (1) that is for (a=q) instead of (a=q) + ; they made the following conjecture: Conjecture 0.0.1 (Murty and Saradha). Let q > 1 be a positive integer and K be an algebraic number eld over which the q-th cyclotomic polynomial is irreducible. Then, the numbers (a=q) where 1 a q 􀀀 1 with (a; q) = 1 are linearly independent over K: In relation to the above conjecture, Gun, Murty and Rath in [28] have also made some improvements concerning the linear independence of digamma values at rational arguments. The central object of study in this thesis is the linear independence of family of numbers over the eld K as well as the eld of algebraic numbers. In this thesis, we study a family of numbers known as harmonic numbers which are de ned as Hn = 1 + 1 2 + + 1 n ; that arises from the truncation of the harmonic series. In fact by using the series representation of the harmonic function it is easy to see that the harmonic function is closely related to the digamma function by the following relation Hr = (r + 1) + ; r 62 Z􀀀: From the previous discussion it is natural to ask the linearly independence of these harmonic numbers over the eld of algebraic numbers. In this thesis we investigate the transcendence nature of harmonic numbers along with the dimension of the space generated by these numbers for some special cases. We mainly focus on the dimension of the space generated by these harmonic numbers and some of the main ingredients that we have used are Baker's theory [1] of linear forms in logarithms of algebraic numbers, theory of multiplicative independence of cyclotomic units [58] and Galois theory to prove the non-singularity of the matrix that we have obtained while solving some linear equations. Next we consider another set of numbers known as cyclotomic numbers and multiplicative independence of these numbers is related to a conjecture of Livingston known as Livingston's conjecture. 1n 1965, in an attempt to solve a conjecture of Erd}os (see [18] and [43]) about the non-vanishing of L- functions associated with certain periodic arithmetic functions, Livingston [43] made another conjecture and predicted that his conjecture is su cient to prove the conjecture of Erd}os. In 2016, Siddhi Pathak [53] disproved the Livingston's conjecture with the help of Dedekind determinant. Along with this the author also gave some necessary conditions under which the conjecture is true. In fact she also observed that Livingston's conjecture is not su cient to prove Erd}os conjecture. Here in this thesis, we give a new proof of the Livingston's conjecture that involves the identities of the sine function at rational arguments and later we modify this conjecture by introducing co-primality condition and in that case we provide the necessary and su cient conditions for the conjecture to be true. As mentioned earlier, we also study the Conjecture 0.0.1 by Murty and Saradha [47] on linear independence of digamma values. Along with the conjecture, the authors [47] have also established a connection between the arithmetic nature of the Euler's constant and the Conjecture 0.0.1. In this thesis, we rst prove that the Conjecture 0.0.1 is true with at most one exceptional q. Later on we also make some remarks on the linear independence of these digamma values with the arithmetic nature of the Euler's constant : Later, we study another conjectures of Gun, Murty and Rath ([28] and [30]) which are regarded as a variant of Rohrlich conjecture and a variant of Rohrlich-Lang conjecture about the linear independence of the logarithms of gamma values over the eld of rationals and algebraic numbers respectively. In this thesis, we provide counter examples to these variants of conjecture of Rohrlich and Lang for an in nite class of integers having at least two prime factors satisfying certain conditions and also make some improvements for the remaining cases. Finally, we study the special values of the derivatives of L-functions attached to certain numbers theoretic constants. In particular, we establish the non-vanishing of the special values of derivatives of L-functions attached to certain even periodic arithmetic functions.