Linear Independence of Logarithms of Cyclotomic Numbers and a Conjecture of Livingston

Abstract

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.

A note on a variant of a conjecture of Rohrlich

“…As a corollary of Theorem 1.13, we show that when 𝑞 ≡ 2(mod 4) and 𝑞∕2 satisfy none of the conditions given in Proposition 2. 3 Finally, as a corollary of the above theorem we have the following theorem: Theorem 1.15. Let 𝑞 > 1 be an integer as in Theorem 1.14.…”

“…The next propositions are of great use as these extend the idea of multiplicative independence of cyclotomic units in Proposition 2.3 for the case when 𝑞 ≡ 2 (mod 4) (for a proof see[3]). …”

“…Remark Note that the above theorem deals with the cases when $$q \not\equiv 2 (\text{ mod } 4)$$. For the case when $$q \equiv 2 (\text{ mod } 4)$$ with $$q=2m$$ where m is an odd integer, then by using the [3, Lemma 2.4], the set $$\begin{equation*} S={\left\lbrace \log {\left(2 \sin \frac{a\pi }{q}\right)}\big\vert \ (a,q)=1, \ 1< a < q/2\right\rbrace} \end{equation*}$$is linearly independent over $${\mathbb {Q}}$$ if and only if the system $$\begin{equation*} T={\left\lbrace \log {\left(2 \sin \frac{h\pi }{m}\right)}\big\vert \ (h,m)=1, \ 1< h < m/2\right\rbrace} \end{equation*}$$is linearly independent over $${\mathbb {Q}}$$ with $$|S|=|T|$$. Thus for the case when $$q \equiv 2 (\text{mod } 4)$$ with $$q=2m$...$…”

“…The next propositions are of great use as these extend the idea of multiplicative independence of cyclotomic units in Proposition 2.3 for the case when $$q \equiv 2 \text{ (mod 4)}$$ (for a proof see [3]). Proposition Let $$q \geqslant 2$$ be a positive integer.…”

“…and 𝐵(1∕3, 1∕2) = √ 3Γ(1∕3) 3 √ 2 4∕3 𝜋 are transcendental in nature. But the presence of 𝜋 in both the above equations makes it difficult to understand the nature of the numbers Γ(1∕4) and Γ(1∕3).…”