Is there a way to analytically extend $x^2+x^3+x^5+x^7+\cdots+x^{p_n}+\cdots$?

Question

Is there a way to analytically extend $x^2+x^3+x^5+x^7+\cdots+x^{p_n}+\cdots\text{?}$

I was wondering because I know that for $|x|$ smaller than $1$ it converges.

I have been wondering for 2 years now I haven't found an answer.

Answer 1

This is part of the theory of pseudocontinuable functions (or generalized analytic continuation theory as presented in the monograph of Ross and Shapiro linked here) and a result of Alexandrov shows that any power series supported on the primes must have the convergence radius higher than $1$ to be pseudo-continuable across any arc of the unit disc (so in particular pseudo-continuation or sometimes called generalized analytic continuation across any (nondegenerate) arc, implies analytic continuation across the full circle).

The result is true for many other sets which are not that sparse like the set of sums of two squares, the set of numbers with at most $k$ prime factors (for fixed $k$ - eg $k=1$ are the primes); also the result holds for power series supported on the set of squares but it is not known afaik for power series supported on the set of cubes for example despite that is fairly sparse.

So the series in the OP is not continuable in even a weak sense anywhere beyond the unit disc.

By definition, a pseudocontinuable power series across a (non-degenerate) arc is a holomorphic function in the unit disc $f(z)$ for which there exists a meromorphic function $F$ in some annulus $1< |z| <R$ which has the same non-tangential limits as $f$ ae on the respective arc (in particular we have $\lim_{r \to 1, r<1}f(re^{it})=\lim_{r \to 1, r<1}F(e^{it}/r)$ almost everywhere for $e^{it} \in J$ the given arc).

A classic example of pseudocontinuation (Poincare) is the series $\sum \frac{c_n}{z-e^{it_n}}, \sum |c_n| < \infty, e^{it_n}$ dense in the unit circle, for which the functions $f(z), |z|<1$ and $F(z), |z|>1$ given by the above are both analytic, have the unit circle as natural boundary (so have no analytic continuation at any point of the unit circle) but satisfy the pseudo-continuability property so form a "coherent" pair in the sense that one can be recovered from the other by the radial-limits.