A note on a variant of a conjecture of Rohrlich

Abstract

Around the late 1970s, Rohrlich made a conjecture about multiplicative algebraic relations among the special values of the Γfunction. Later, Lang generalized the Rohrlich conjecture to polynomial algebraic relations among special values of the gamma function. In 2009, Gun et al. (J. Number Theory 129 (2009),no. 8, 1858-1873) formulated a variant of this conjecture of Rohrlich and a variant of the conjecture of Lang that deals with the linear independence of the values at non-integeral rational numbers of the logarithm of the gamma function over the field of rationals and algebraic numbers, respectively. In this direction, they proved a set of interesting results for the case of primes and their powers over the field of rationals. Further for the case of prime powers, they have extended their results assuming the Schanuel's conjecture. In this article, we improve their results without assuming Schanuel's conjecture. Further we provide counter examples to these variants of conjectures of Rohrlich and Lang for an infinite class of integers having at least two prime factors satisfying certain conditions. M S C ( 2 0 2 0 )11J81, 11J86 (primary), 11J91 (secondary)