I'm interested in information about the power series
$$\sum_{\text{$p$ prime}} x^p$$
and the related power series
$$\sum_{n=1}^\infty (-1)^n x^{p(n)}$$
where $p(n)$ is the nth prime.
Immediately, the gap theorem shows that these series have a natural boundary at the unit circle. However, I'm especially interested in any attempts to provide a (non-analytic) continuation of these functions. Have these functions been studied before?
Best Answer
Indeed these functions support a large family of lambert series based non analytic continuations. Around $0$ we have that
$$ \sum_{n=1}^{\infty} x^p = x^2 + \frac{x^3}{1-x^2} - \frac{x^9 +x^{15} + x^{21}+x^{25}+x^{27}}{1-x^{30}} - \frac{x^{33} + \ ... x^{203}}{1-x^{210}} ...$$
Where $2,6,30,210...$ as the primorials and the numerators i'll explain below but basically correspond to a non eratosthenian sieve. This series is curiously well defined outside of the unit circle.
There's actually infinitely many such forms. Another example is given by:
$$ \sum_{n=1}^{\infty} x^p = x^2 + \frac{x^3}{1-x^2} - \frac{x^9 + x^{15} + x^{21}}{1-x^{24}} - \frac{x^{25} + ... x^{115}}{1-x^{120}} ...$$
Where $2,6,24,120...$ are the factorials and the numerators are generated a similar way. What's surprising is that, these two forms also appear to be very similar (possibly equal once all terms are included, though I am not sure) outside the natural boundary.
(For some clarification I haven't graphed the entire numerator because there are a lot of terms. A link to the graph is here):
Explanation of the Numerators:
We start by considering the most naive strategy which is just implementing the sieve of eratosthenes via a lambert series. We should have:
$$ \sum_{n=1}^{\infty} x^p = \frac{x^2}{1-x} - \frac{x^4}{1-x^2} - \frac{x^9}{1-x^3} + \frac{x^{12}}{1-x^6} - \frac{x^{25}}{1-x^5} + ... $$
Where there is an inclusion exclusion occurring depending on how many factors a particular number has. The trouble with the Sieve of Eratosthenes is that the numerators all have higher polynomial order than the denominators so this cannot possibly converge for $|x|>1$.
But the sieve of Eratosthenes isn't the only way to sieve for primes. We have infinitely many congruences we haven't used. For example we know that there are no prime numbers which are $0 \mod 4$ and no prime numbers which are $4 \mod 6$ so on and so forth.
So a different strategy is to consider a sequence of natural numbers $b_1, b_2, b_3,... $ where for any $n$ there exist infinitely many $m>n$ such that $b_n | b_m$ and that for every prime $p$ there exists some $b_k$ such that $p|b_k$. Given such a sequence $b_i$ we can then generate a set of congruences which exclude everything but the primes. We consider the primorial examples
$$ b_1 = 2, b_2 = 2\times 3= 6, b_3 = 2\times 3 \times 5 = 30, b_4 = 2\times 3 \times 5 \times 7 = 210 ... $$
We then begin generating congruences.
$$ \begin{matrix} p\ne 4 \mod 6 \\ p \ne 4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,27,28 \mod 30 \\ \vdots \end{matrix} $$
etc...
Now we perform a filter if any congruence $a \ne b \mod c$ is implied naturally by $d \ne e \mod f$ where $f<c$ then we can ignore the congruence $a \ne b \mod c$ since its already been accounted for.
To save myself effort I used the term $x^2 + \frac{x^3}{1-x^2}$ which only has odd terms so I remove those even congruences and likewise at the $\mod 210$ layer I remove any congruences implied by the $\mod 30$ layer
$$ \begin{matrix} \emptyset \\ p \ne 9,15,21,25,27 \mod 30 \\ p \ne 33, 35,49, 63, 65, 77, 91, 93, 95, 119, 123, 125, 133, 153, 155, 161, 183, 185, 203 \mod 210 \\ \vdots \end{matrix} $$
And this results in the numerators in the first example.
Why one should be suspicious
Lambert series like this can lead to non standard continuations. The book generalized analytic continuation by Ross, and Shapiro explores these ideas and gives an example in page 1 of chapter 1 of a lambert series that is $\frac{x}{1-x}$ inside the unit disk and $-\frac{x}{1-x}$ outside the unit disk reproduced below:
$$ \frac{x}{1-x^2} + \frac{x^2}{1-x^4} + ... + \frac{x^{2^{n-1}}}{1-x^{2^n}} + ... $$
But even then I think there is a space of ideas here worth exploring. I also have examples (In particular cubic theta series via sieving) where the standard continuation seems like the only possible continuation via lambert series.
At the boundary:
Evaluating $f(z) = \sum_{k=0}^{\infty} x^p$ at the boundary is intimately tied to prime number races. For example $f(i) = -1 -i + i -i -i ... $ where the terms strictly oscillate between $i, -i$ depending on if a prime is $1 \mod 4$ or $3 \mod 4$. Our Characterizion of prime number races is (to my amateur knowledge) not accurate enough to be able to assign these divergent sums.