In 1965, A. Livingston conjectured the $\overline{\mathbb{Q}}$-linear independence of logarithms of values of the sine function at rational arguments. In 2016, S. Pathak disproved the conjecture. In this article, we give a new proof of Livingston’s conjecture using some fundamental trigonometric identities. Moreover, we show that a stronger version of her theorem is true. In fact, we modify this conjecture by introducing a co-primality condition, and in that case we provide the necessary and sufficient conditions for the conjecture to be true. Finally, we identify a maximal linearly independent subset of the numbers considered in Livingston’s conjecture.

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Let H n = n k=1 1 k be the n-th harmonic number. Euler extended it to complex arguments and defined H r for any complex number r except for the negative integers. In this paper, we give a new proof of the transcendental nature of H r for rational r. For some special values of q > 1, we give an upper bound for the number of linearly independent harmonic numbers H a/q with 1 ≤ a ≤ q over the field of algebraic numbers. Also, for any finite set of odd primes J with |J| = n, defineFinally, we show that

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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)