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A very simple question, but I can't seem to find anything relating to it :

Is there any research, are there any results that have focused on or given insight on

$\sum 1/p^p$, ${p \in \mathbb P}$ ?

A very basic series, converges extremely fast, its value is around .29. What more can there be said about it ?

From what little I know about more advanced number theory, similar sequences (I can think of a few similar ones that I can't find any relevant research or results about) can be very non-trivial to compute or to analyse.

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  • $\begingroup$ The first digits are at the Prime pages and appear too here and here and the question was asked earlier at SE. $\endgroup$ – Raymond Manzoni Nov 15 '15 at 14:22
  • $\begingroup$ Thank you ! I'm surprised I didn't find that topic when searching for it. $\endgroup$ – spliblib Nov 15 '15 at 14:30
  • $\begingroup$ How can we search "1/p^p" here? I don't know. $\endgroup$ – GEdgar Nov 15 '15 at 14:33
  • $\begingroup$ @GEdgar and spliblib: I searched the numerical value using google (reducing progressively the number of digits since only exact matches will return results). $\endgroup$ – Raymond Manzoni Nov 15 '15 at 15:41
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This is OEIS A094289, where they have no information except computations of the value. This suggests the answer "no" to the question "is there any research ..."

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  • $\begingroup$ Ah, I guess that answers it, then. Thank you. $\endgroup$ – spliblib Nov 15 '15 at 14:28
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What more can there be said about it ?

Essentially nothing. Related series are

No closed form expressions for these constants are known so far.

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  • $\begingroup$ I am not asking specifically for closed form expressions. I am aware of the fact that similar expressions have no closed forms. What I meant is that I have found nothing whatsoever relating to the sequence I mentioned, whereas the Sophomore series has a couple of things that can be said about it (the formula in the wikipedia page, links to Stirling's formula), and the prime zeta series seems to have been extensively studied. $\endgroup$ – spliblib Nov 15 '15 at 11:35
  • $\begingroup$ @spliblib This is why I give the link to prime zeta function. It seems to me to be the simplest reasonable relative of your series. $\endgroup$ – Start wearing purple Nov 15 '15 at 11:36
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it may not be elegant but as an idea $\sum = \sum_{n=1}^{n=5}1/{p^p} + \sum_{n=6}1/{p^p}$ and as per this leverage approximation / boundaries

$\log{n} + \log{\log{n}} - 1 < \frac{p_n}{n} < \log{n} + \log{\log{n}}$ for $n \geq 6$

and then look at convergence of $\sum_{n=6}$

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Other than its rational approximation

$$ \frac{5226294}{18187381}, $$

not much is known about this constant. In particular, its irrationality/transcendentality seems to be out of reach with current technology. However, it can be related to similar constants, such as:

  • The prime zeta function: $$ P(k)=\sum_{p\in\mathbb P}\frac{1}{p^k}, \ \ Re(k)>1, $$

  • $n$-ary representations of the Prime Constant: $$ f(k)=\sum_{p\in\mathbb P}\frac{1}{k^p}, $$

(noting that $f(2)$ is the Prime Constant $\rho=0.414682509\ldots$),

  • Sophomore's constant:

$$ C_s=\sum_{n=1}^{\infty}\frac{1}{n^n}. $$

As pointed out by others, there is more information on the OEIS website (A094289).

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