I was curious to know what the following limit is:
$$\lim_{x\downarrow-1}\sum_{n=1}^\infty p_nx^{n-1}=\lim_{x\downarrow-1}(2+3x+5x^2+7x^3+11x^4+\dots)$$
where $p_n$ is the $n$th prime.
I graphed the first 6 or so partial sums:
but they converge terribly slow. WolframAlpha doesn't seem to have much of a clue either, and with some numerical testing, it seems this limit might go to $\infty$, which is quite strange, since it suggests even and odd indexed primes are not asymptotically equally spaced.
By using some asymptotes on the growth rate of the $n$th prime, one can also easily deduce that
$$\sum_{n=1}^\infty p_nx^{n-1}$$
converges absolutely for $|x|<1$.
Can we prove this limit goes off to $\infty$ or that it exists? And if it does exist, what is it's value?
Since $p_n=\Theta(n\log n)$ by the weak version of the PNT, we may compute the wanted limit through a convolution with an approximate identity:
$$ \lim_{x\to 0^+} \sum_{n\geq 1}p_n(x-1)^{n-1} = \lim_{n\to +\infty} \sum_{n\geq 1}p_n \int_{0}^{+\infty}n(x-1)^{n-1} e^{-nx}\,dx $$ where $$ \int_{1}^{+\infty}n(x-1)^{n-1}e^{-nx}\,dx = \frac{n!}{e^n n^n}\approx\frac{\sqrt{2\pi n}}{e^{2n}}$$ gives no issues, but $$ \int_{0}^{1}n(x-1)^{n-1} e^{-nx}\,dx=(-1)^{n-1}\int_{0}^{n}\left(1-\frac{x} {n}\right)^{n-1}e^{-x}\,dx $$ behaves like $\frac{(-1)^n}{2}$ for large values of $n$. In particular the existence of the wanted limit depends on the Cesàro summability of the sequence $\{(-1)^n p_n\}_{n\geq 1}$, hence on the distribution of prime gaps.
The result of Ping Ngai Chung and Shiyu Li mentioned on MO implies that the wanted limit does not exist.