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The sum over primes:

$$\sum_{k=0}^\infty \{\text{k if k is prime, 0 otherwise}\}$$

I know that there is no known method to ascribe a reasonable value to the sum of the primes https://www.quora.com/What-is-the-sum-of-all-primes but what about a case when the primes are spaced with zeros in place of non-primes?

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  • $\begingroup$ This question is not as trivial as it superficially looks, so in my opinion the downvotes and closing votes were hasty. However, you should explain what regularization you want. $\endgroup$
    – Jan-Christoph Schlage-Puchta
    Commented May 25, 2018 at 7:38
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    $\begingroup$ A naive attempt of regularization could be considering the function $F(s)=\sum_{p} p^{-s}$ and evaluating $F(-1)$ for its analytical continuation (if it exists). $\endgroup$
    – Fedor Petrov
    Commented May 25, 2018 at 8:20
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    $\begingroup$ @FedorPetrov Unfortunately, $F(s)$ (known as prime zeta function) cannot be analytically continued that far. It can be extended arbitrarily close to the imaginary axis, but there are logarithmic singularities clustering next to it, turning the axis into a natural boundary - there is no connected domain crossing the axis on which $F(s)$ can be defined. (edit: just realized this is already said in the Quora answer OP links) $\endgroup$
    – Wojowu
    Commented May 25, 2018 at 12:54
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    $\begingroup$ One more note: the prime zeta function is the function which adds up primes with zeros inbetween, since it is a Dirichlet series $\sum_n I_{\mathbb P}(n)/n^s$. Hence the Quora answer you link actually answers your question. It's less clear how we would go around to regularizing the sum $\sum_np_n$ - one idea is to consider Dirichlet series $\sum_n p_n/n^s$, but I don't think this function can be (easily) extended to the right of $\sigma=2$, let alone to $s=0$. $\endgroup$
    – Wojowu
    Commented May 27, 2018 at 21:27
  • $\begingroup$ To be frank, I am already sick of this sort of questions. Regularizing a divergent series is a "problem" which makes no sense whatsoever without additional information. The most common way to make it reasonable is to consider a given series a special case of a family, but then, it makes a huge difference how exactly you do this. $\endgroup$
    – Alex Gavrilov
    Commented Jun 2, 2018 at 14:15

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