- It is not hard to construct a Dirichlet series $$f(z)=\sum_{n=1}^\infty a_ne^{\lambda_n z}$$ which converges to $0$ absolutely and uniformly on the real line but does not converge in the complex plane. It is constructed as a sum of 3 series $f=f_0+f_1+f_2.$ Let $f_1$ be a series with imaginary exponents $\lambda_n$ which converges to an entire function in the closed lower half-plane, but not in the whole plane. Such series is not difficult to construct, see V. Bernstein, page 34, (see the full reference below) and there are simpler examples, with ordinary Dirichlet series. Then put $f_2=\overline{f_1(\overline{z})}$, and $f_0=-f_1-f_2$. So all three functions are entire. Now, according to Leontiev, EVERY entire function can be represented by a Dirichlet series which converges in the whole plane. Thus we have a Dirichlet series $f_0+f_1+f_2$ which converges on the real line to $0$ but does not converge in the plane.
I. It is not hard to construct a Dirichlet series $$f(z)=\sum_{n=1}^\infty a_ne^{\lambda_n z}$$ which converges to $0$ absolutely and uniformly on the real line but does not converge at some points of the complex plane. It is constructed as a sum of 3 series $f=f_0+f_1+f_2.$ Let $f_1$ be a series with imaginary exponents $\lambda_n$ which converges to an entire function in the closed lower half-plane, but not in the whole plane. Such series is not difficult to construct, see V. Bernstein, page 34, (see the full reference below) and there are simpler examples, with ordinary Dirichlet series. Then put $f_2=\overline{f_1(\overline{z})}$, and $f_0=-f_1-f_2$. So all three functions are entire. Now, according to Leontiev, EVERY entire function can be represented by a Dirichlet series which converges in the whole plane. Thus we have a Dirichlet series $f_0+f_1+f_2$ which converges on the real line to $0$ but does not converge in the plane.
- It is clear from the work of Leontiev, that to obtain a reasonable theory, one has to restrict to exponents of finite upper density, $n=O(|\lambda_n|)$, otherwise there is no uniqueness in $C$. In the result I cited above the expansion of $f_0$ is highly non-unique.
But thereII. It is one more reasonable conditionclear from the work of Leontiev, that the $\lambda_n$ cannot come too close together.to obtain a reasonable theory, This is called "finite indexone has to restrict to exponents of concentration" in Bernstein's book. When you have a finite index of concentrationupper density, $f_1$ and$n=O(|\lambda_n|)$, otherwise there is no uniqueness in $f_2$ as above do not exist:$C$. In the result I cited above the expansion of $f_0$ is highly non-unique.
Assuming finite upper density I proved that if a Dirichlet series is ABSOLUTELY and uniformly (with say imaginary exponents) convergesconvergent on the real line to an entire function in a half-planezero, then it convergesall coefficients must be zero. everywherehttp://www.math.purdue.edu/~eremenko/dvi/exp2.pdf I don't know how to get rid of the assumption of absolute convergence.
- Assuming finite upper density AND finite index of concentration, I believe that I can prove $R$-linear independence. In fact a stronger property: if a Dirichlet series (with complex coefficients) converges on $R$ to $0$, then it has all zero coefficients.
But there is a philosophical argument in favor of absolute convergence: the notion of "linear dependence" should not depend on the ordering of vectors:-)
III. The ideamost satisfactory result on my opinion, is that of Schwartz. Let us say that the exponentials are S-linearly independent if none of them belongs to the proof is followingclosure of linear span of the rest. SplitTopology of uniform convergence on compact subsets of the exponents into three setsreal line. Schwartz gave a necessary and sufficient conditon of this: a) those inthe points $|\arg(z-\pi/2)|<\pi/10,$ b) those$i\lambda_k$ must be contained in $|\arg(z+\pi/2)|<\pi/10,$ and the rest. Sozero-set of the series is split asFourier transform of a measure with a bounded support in R.
(L. Schwartz, Theorie generale des fonctions moyenne-periodiques, Ann. Math. 48 $f_1+f_2+f_3$(1947) 867-929.)
Now itA complete explicit characterization of such sets is easy to prove that if $f_3$ converges at all real pointsnot known, then it converges uniformly andbut they have finite upper density, abcolutely on all compacts in the planeand many of their properties are understood. And thus represents anThese Fourier transforms are entire functionfunctions of exponenitial type bounded on the real line. The link I gave above contains Schwartz's proof Itin English. S-linear dependence is also easynon-sensitive to prove that for $f_1,f_2$ convergence at all real points implies uniform and absolute convergence in the closed upper and lower half-planes, respectively. So we have 3ordering of functions, one entire one analytic in the lower halfplane and continuous on the closed lower half-planewhich is good.
IV. Vladimir Bernstein's book is "Lecons sur les progress recent de la theorie des series de Dirichlet", and the last oneParis 1933. This is analytic in the upper half-planemost comprehensive book on Dirichlet series, continuous in the closure.but unfortunately And on theonly with real line their sum is 0exponents.
It follows (from the removable singularity theorem) that all three are entireV. If The application to the indexfunctional equation mentioned by the author of concentrationthe problem is finite this would imply thatnot a good justification for the seriesstudy of $f_1$ and $f_2$ converge in the whole planeproblem in such generality. By uniququeness the sumThe set of exponentials there is zero in the whole plane very simple, and certainly we are donehave $R$-linear independence for SUCH set of exponentials. Besides the theorem stated as an application has been proved in an elementary way.
This is only a sketch becauseVI. Finally, I recommend to change the statement that finite indexdefinition of concentration implies convergence$R$-linear independence by allowing complex in the whole plane is provedcoefficients (by Bernsteinbut equality to $0$ on the real line) only for imaginary exponents. However I believe that I can extend this toAgain in the case of exponentsapplication mentioned in a small sector around the imaginary axisoriginal problem, THIS notion of $R$-uniqueness is needed: the function is real, but the exponentials are not real, thus coefficients should not be real.
Vladimir Bernstein's book is "Lecons sur les progress recent de la theorie des series de Dirichlet", Paris 1933. This is the most comprehensive book on Dirichlet series, but unfortunately only with real exponents.
The application to the functional equation mentioned by the author of the problem is not a good justification for the study of the problem in such generality. The set of exponentials there is very simple, and certainly we have $R$-linear independence for SUCH set of exponentials. Besides the theorem stated as an application has been proved in an elementary way.
If the exponents $\lambda_k$ are zeros of Fourier transform of a measure with bounded support on the real line, we have $R$-linear independence. This result is sufficient for treating all applications to convolution equations as in the example in the original problem. This is due to L. Schwartz, Theorie generale des fonctions moyenne-periodiques, Ann. Math. 48 (1947) 867-929.
Finally, I recommend to change the definition of $R$-linear independence by allowing complex coefficients (but equality to $0$ on the real line). Again in the application mentioned in the original problem, THIS notion of $R$-uniqueness is needed: the function is real, but the exponentials are not real, thus coefficients should not be real.
In the same paper Schwartz proves that if $\lambda_n$ are NOT zeros of a Fourier transform of a measure with bounded support, then one expoiential, say $e^{\lambda_0z}$ can be approximated by finite linear combinations of the others uniformly on compact subsets. This does not give a literal "linear dependence" yet, because the resulting series converges only after a grouping of some terms.