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It is possible, by means of zeta function regularization and the Ramanujan summation method, to assign a finite value to the sum of the natural numbers (here $n \to \infty $) :

$ 1 + 2 + 3 + 4 + \cdots + n \; {“ \;=\; ”} - \frac{1}{12} . $

Is it also possible to assign a value to the sum of primes, $ 2 + 3 + 5 + 7 + 11 + \cdots + p_{n} $ ($n \to \infty$) by using any summation method for divergent series?

This question is inspired by a question on quora.

Thanks in advance,

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    @Max: Note that $p_n\sim n\log n$ by the prime number theorem. You can zeta-regularize the divergent sum $\sum_{n=1}^\infty n\log n$ by evaluating $-\zeta'(-1)=\log A-1/12$, where A is the Glaisher-Kinkelin constant. So it's an answer to something similar to your question.2011-11-22

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Fröberg shows in his paper that the prime zeta function

$P(s)=\sum_{p\in \mathbb P} \frac1{p^s}=\sum_{k=1}^\infty \frac{\mu(k)}{k}\log\zeta(ks)$

where $\mu(k)$ and $\zeta(s)$ are respectively the Möbius and Riemann functions, cannot be analytically continued to the left half-plane, $\Re\,s\leq 0$ (in particular, we cannot give a reasonable evaluation of $P(-1)$), due to the clustering of poles along the imaginary axis arising from the nontrivial zeros of the Riemann $\zeta$ function.

prime zeta plots

Note the nasty-looking left edges in both plots above.

This result is originally due to Landau and Walfisz. See the linked papers for more details.

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    Thanks, Eri$c$ and anon, for the added details.2011-11-22