I am frequently asked whether exponentials are "linearly independent". That is if we have a sequence of distinct complex numbers $\lambda_j$, whether $$\sum_{j} a_j\exp(\lambda_j z)\equiv 0$$ implies that all $a_j=0$. If the linear dependence above holds in all complex plane, this question was answered by A. F. Leontjev. All his books exist only in Russian, but I was asked this question so frequently that I wrote a short note, Linear independence of exponentials, explaining Leontjev's results, and posted it on Internet.

Then a question was asked on MO, On linear independence of exponentials, what if the linear dependence relation holds only for real $z$ (the $\lambda_j$ and $a_j$ are still complex). I could not find an answer in Leontjev's papers, and the result was a preprint http://www.math.purdue.edu/~eremenko/dvi/exp2.pdf

I am frequently asked whether exponentials are "linearly independent". That is if we have a sequence of distinct complex numbers $\lambda_j$, whether $$\sum_{j} a_j\exp(\lambda_j z)\equiv 0$$ implies that all $a_j=0$. If the linear dependence above holds in all complex plane, this question was answered by A. F. Leontjev. All his books exist only in Russian, but I was asked this question so frequently that I wrote a short note, Linear independence of exponentials, explaining Leontjev's results, and posted it on Internet.

Then a question was asked on MO, On linear independence of exponentials, what if the linear dependence relation holds only for real $z$ (the $\lambda_j$ and $a_j$ are still complex). I could not find an answer in Leontjev's papers, and the result was a preprint https://www.math.purdue.edu/~eremenko/dvi/exp2.pdf

I am frequently asked whether exponentials are "linearly independent". That is if we have a sequence of distinct complex numbers $\lambda_j$, whether $$\sum_{j} a_j\exp(\lambda_j z)\equiv 0$$ implies that all $a_j=0$. If the linear dependence above holds in all complex plane, this question was answered by A. F. Leontjev. All his books exist only in Russian, but I was asked this question so frequently that I wrote a short note, Linear independence of exponentials, explaining Leontjev's results, and posted it on Internet.

Then a question was asked on MO, On linear independence of exponentials, what if the linear dependence relation holds only for real $z$ (the $\lambda_j$ and $a_j$ are still complex). I could not find an answer in Leontjev's papers, and the result was a preprint http://www.math.purdue.edu/~eremenko/dvi/exp2.pdf

I am frequently asked whether exponentials are "linearly independent". That is if we have a sequence of distinct complex numbers $\lambda_j$, whether $$\sum_{j} a_j\exp(\lambda_j z)\equiv 0$$ implies that all $a_j=0$. If the linear dependence above holds in all complex plane, this question was answered by A. F. Leontjev. All his books exist only in Russian, but I was asked this question so frequently that I wrote a short note, Linear independence of exponentials, explaining Leontjev's results, and posted it on Internet.

Then a question was asked on MO, On linear independence of exponentials, what if the linear dependence relation holds only for real $z$ (the $\lambda_j$ and $a_j$ are still complex). I could not find an answer in Leontjev's papers, and the result was a preprint http://www.math.purdue.edu/~eremenko/dvi/exp2.pdf