02/08/2023, 05:28 AM
I haven't read the whole post yet in detail (I'd like to take some time to think about the ideas you bring up), but as quick note, let me say a few things. First, the natural boundary I was talking about is the dense set of singularities definition-- a line at which the function fails to be able to be analytically continued since the singularities are so plentiful there is no path through them (i.e. in every open interval on the line of any size there is a singularity).
I haven't gone through and proved the function is not meromorphic, but, in case you didn't already know-- "most" functions have a natural boundary! For instance, if you pick a random Taylor series, it will have natural boundary at the unit circle (with probability 1). A similar result holds for Dirichlet series, if you pick one with random coefficents it will have a natural boundary on some line. In fact, (Polya?) has a theorem that has changed my view forever. If you pick a power series with integer coeffiencts, it is either a rational function or has a natural boundary! This is pretty absurd-- my own 'corollary' is that "A power series with integer coeffiecents is either interesting or doesn't have a natural boundary." Nowadays, whenever I see a function with lots of singularities, I pretty much just automatically assume its got a natural boundary.
However, I do actually have some reasons to believe Gottfried's function has a natural boundary. Consider instead the series
\[F_x(z)= \sum (-1)^n z^{n^x}\]
I write the series with x as a subscript to emphasize that I fix x and think of z as varying. For integer x>1 this function actually has a natural boundary on the unit circle, but something called the Fabry gap theorem, which says that when the coeffiecents are integers growing infinitely faster than \(n\) you end up with a function with a natural boundary. Notice that at x=2 \( F_2(z)\) is basically the theta function, so its pretty much a modular form. One also has that
\[G_x(z)= \sum (-1)^n z^{\lfloor (n^x) \rfloor}\]
has a natural boundary for all real numbers x>1. Even without the floor being applied, I'd say its a safe bet that it should still have a natural boundary. Basically, if the coeffiecnts grow too fast, you should expect it to have a natural boundary. Now, Gottfrieds function
\[ \sum (-1)^n n^{n^x}\]
grows even faster, and than the other functions, so it probably should have a natural boundary. In fact, the line \( \mathfrak{R}(x)>1 \) is right where the growth of the coefficents graduates beyond even factorial growth-- it becomes too fast growing for Borel transformations to work. I've never heard of a function with such fast growth rate that doesn't have a natural boundary-- I don't know if such a thing is even possible. So anyway, Gottfried's function probably has a natural boundary, or at least, I would be highly suprised if it didn't have a boundary there.
I haven't gone through and proved the function is not meromorphic, but, in case you didn't already know-- "most" functions have a natural boundary! For instance, if you pick a random Taylor series, it will have natural boundary at the unit circle (with probability 1). A similar result holds for Dirichlet series, if you pick one with random coefficents it will have a natural boundary on some line. In fact, (Polya?) has a theorem that has changed my view forever. If you pick a power series with integer coeffiencts, it is either a rational function or has a natural boundary! This is pretty absurd-- my own 'corollary' is that "A power series with integer coeffiecents is either interesting or doesn't have a natural boundary." Nowadays, whenever I see a function with lots of singularities, I pretty much just automatically assume its got a natural boundary.
However, I do actually have some reasons to believe Gottfried's function has a natural boundary. Consider instead the series
\[F_x(z)= \sum (-1)^n z^{n^x}\]
I write the series with x as a subscript to emphasize that I fix x and think of z as varying. For integer x>1 this function actually has a natural boundary on the unit circle, but something called the Fabry gap theorem, which says that when the coeffiecents are integers growing infinitely faster than \(n\) you end up with a function with a natural boundary. Notice that at x=2 \( F_2(z)\) is basically the theta function, so its pretty much a modular form. One also has that
\[G_x(z)= \sum (-1)^n z^{\lfloor (n^x) \rfloor}\]
has a natural boundary for all real numbers x>1. Even without the floor being applied, I'd say its a safe bet that it should still have a natural boundary. Basically, if the coeffiecnts grow too fast, you should expect it to have a natural boundary. Now, Gottfrieds function
\[ \sum (-1)^n n^{n^x}\]
grows even faster, and than the other functions, so it probably should have a natural boundary. In fact, the line \( \mathfrak{R}(x)>1 \) is right where the growth of the coefficents graduates beyond even factorial growth-- it becomes too fast growing for Borel transformations to work. I've never heard of a function with such fast growth rate that doesn't have a natural boundary-- I don't know if such a thing is even possible. So anyway, Gottfried's function probably has a natural boundary, or at least, I would be highly suprised if it didn't have a boundary there.
