Is the function $P_m(x) = 2^mx + 3^mx^2 + ... +p_n^mx^n +...$ mathematically useful? Does it converge to known constants for rational m and x? Note that $p_n$ denotes the nth prime number.
I know that power series are a powerful method for obtaining information about sequences, but the only mention of this function that I've come across thus far in my studies is Euler's proof that $P_{-1}(1)$ diverges. From my naive perspective this function seems like a perfectly natural thing to study, and the absence of any mention of it thus far causes me to question my intuition. That is, it causes me to wonder if this function is mathematically useful. Perhaps someone can explain why or why not this function would be a useful thing to study, and if it is useful, cite examples of its use.
The second part of my question is about the convergence of this function. Given a rational value of m, does there exist a rational value of x such that $P_m(x)$ is equal to a known mathematical constant.
Here's a (very possibly false) example to highlight what I mean,
$$P_{-2}(\frac{1}2) = \frac{1}{2^2}\frac{1}2 + \frac{1}{3^2}\frac{1}{2^2} + \frac{1}{5^2}\frac{1}{2^3} + ... + \frac{1}{p_n^2}\frac{1}{2^n} + ... = \alpha\pi$$ for some rational number $\alpha$.
I feel that the questions I tend to ask about power series in comparison to the way they are presented on this site and in textbooks indicates that I am lacking fundamental knowledge about their properties and about what they are "allowed to do" and therefore any answers to this second question are bound to increase my knowledge. I really hope these two questions of mine are clear. I am at an early stage in my mathematical education and I haven't really talked to any professor's yet so I will understand if these questions are perceived as "ill-formed."
 A: This first part will be to answer the convergence questions. Using an upper bound from the Prime Number Theorem and Rosser's Theorem, namely $p_n < n\ln n +n\ln\ln n$ for $n \geq 6$, we can set an easy to work with upper bound $p_n<n^2$ just to get some basic concepts out of the way. Therefore, we can consider the series
$$\sum_{n=1}^\infty {n^{2m}}{x^n}$$
to be greater than your prime series. This series converges for $-1<x<1$ regardless of what $m$ is. At $x=-1$, $m<0$ for the series to converge by the Alternating Series Test. The last case to consider, $x=1$, is the hardest, because we can't really use our upper bound of $p_n<n^2$ anymore. It also gets a little weird, because $\lim_{x\to\infty}x^n>\log(x)\forall n>0$. In other words, a polynomial function will always eventually be greater than a logarithmic function, but a logarithmic function will always eventually be greater than a constant function. Theoretically, any function $x^{1+n}$ will be an upper bound to $x\ln x$, no matter how small the $n$. We can use the series upper bound of $\sum_{n=1}^\infty {n^{(1+n)m}}$. This series will only converge if $(1+n)m<-1$, and, since $n>0$, the only way for this equation to be satisfied is for $m<-1$. In summary, your series should converge for all $m$ when $-1<x<1$, for $m<0$ when $x=-1$, and for $m<1$ when $x=1$.
This part will deal with the convergence of specific values. Basically, all power series that can converge can converge to any specific value within a specific range. For instance, the Maclaurin Series for $e^x$ can converge to all positive numbers, but no negative numbers. For this function, it can converge to all numbers, as setting $m=0$ yields the generic geometric series, which can converge to any number.
Finally, this part will deal with the use of this function. It is important to realize that this function will only be useful when the actual primes do not need to be found or there is no other function that can model a specific behavior and therefore it must be used (i.e. if someone found a way to encode things using your series that worked better than other encryptions). For the Euler example, both of these were true.
