Mathematics – Number Theory
Scientific paper
2009-08-22
Mathematics
Number Theory
8 pages, 1 figure. Submitted in revised form to Acta Aritmetica (12/16/2009)
Scientific paper
\quad In a very recent work [JNT \textbf{129}, 2154 (2009)], Gun and co-workers have claimed that the number $ \log{\Gamma(x)} + \log{\Gamma(1-x)} $, $x$ being a rational number between 0 and 1, is transcendental with at most \emph{one} possible exception, but the proof presented there in that work is \emph{incorrect}. Here in this paper, I point out the mistake they committed and I present a theorem that establishes the transcendence of those numbers with at most \emph{two} possible exceptions. As a consequence, I make use of the reflection property of this function to establish a criteria for the transcendence of $ \log{\pi}$, a number whose irrationality is not proved yet. I also show that each pair $\{\log{[\pi/\sin(\pi x)]}, \log{[\pi/\sin(\pi y)]}\}$, $x$ and $y$ being rational numbers between 0 and 1, contains at least one transcendental number. This has an interesting consequence for the transcendence of the product $ \pi \cdot e$, another number whose irrationality is not proved.
No associations
LandOfFree
On the possible exceptions for the transcendence of the log-gamma function at rational entries does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the possible exceptions for the transcendence of the log-gamma function at rational entries, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the possible exceptions for the transcendence of the log-gamma function at rational entries will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-190542