| In his two well known 1968 papers \og Contributions to the theory of transcendental numbers\fg, K. Ramachandra proved several results showing that, in certain explicit sets $\{x_1,\ldots,x_n\}$ of complex numbers, one element at least is transcendental. In specific cases the number $n$ of elements in the set was $2$ and the two numbers $x_1$, $x_2$ were both real. He then noticed that the conclusion is equivalent to saying that the complex number $x_1+ix_2$ is transcendental. In his 2004 paper published in the Journal de Théorie des Nombres de Bordeaux, G.~Diaz investigates how complex conjugation can be used for the transcendence study of the values of the exponential function. For instance, if $\log \alpha_1$ and $\log \alpha_2$ are two nonzero logarithms of algebraic numbers, one of them being either real of purely imaginary, and not the other, then the product $(\log \alpha_1)(\log \alpha_2)$ is transcendental. We will survey Diaz's results and produce further similar ones. |
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