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Wikipedia has a page about the prime zeta function which is defined as follows:

$$P(s)=\sum_{p\;\text{prime}} \frac1{p^s}$$

I entered this additional definition:

Define a sequence: $$a_n=\prod_{d\mid n} \frac{\Lambda(d)}{\log(d)},$$ where zeros are not included in the multiplication and $a_1=1$ then: $$P(s)=\log\sum_{n=1}^\infty \frac{a_n}{n^s}.$$ Is it a problem that this later definition does not include the zeros in the multiplication?

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    What is $\Lambda(n/d)$? Is it the Von Mangoldt function (http://en.wikipedia.org/wiki/Von_Mangoldt_function)? Then that should be explained both here and at Wikipedia.2011-04-14
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    I might have made a mistake but what I had in mind was the Mangoldt function applied to the divisors of n. In oeis notation which I understand better I would have said: For row>1: a(n)^-1 = row products of A100995(A126988).2011-04-14
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    @joriki Ok, nothing new then. But I don't think this is mentioned in the oeis.2011-04-14
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    What do you mean by "nothing new"? I only linked to the Von Mangoldt function to make what you wrote intelligible to someone who doesn't know this symbol (like I didn't), not to imply that there is nothing new in what you wrote. As you may be aware, it's Wikipedia policy that you shouldn't put anything new in there (http://en.wikipedia.org/wiki/WP:OR). I personally don't think this should apply too rigorously to mathematics, since proofs can be checked without secondary sources. But what you wrote doesn't form a natural part of the article. I think you should either flesh it out or delete it.2011-04-14
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    It's an interesting relationship, by the way. But why don't you write $d$ instead of $n/d$?2011-04-14
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    I guess you are right about the n/d, I should have written only d. I have had problems with this notation because I always think of row index divided by column index when I see it.2011-04-14
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    Your definition of $a_n$ seems rather complicated, all the more so since you have to exclude the zeroes and the Von Mangoldt function isn't terribly well-known -- wouldn't it be more straightforward to just say that $a_n$ contains a factor $1/k!$ for each maximal prime power $p^k$ in its factorization? (Or equivalently a factor $1/k$ for each prime power $p^k$ that divides it.)2011-04-14
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    Also, I wouldn't call what you wrote an "additional definition" of $P$ -- the actual definition is clearly more direct and simple; what you wrote is a relationship involving $P$, not something one would use to define $P$.2011-04-14
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    Mats, I edited the wikipedia page to include Joriki's suggested simplification. I would like to see you add a justification or citation for this identity, otherwise it might be deleted as original research (it might be anyway, but it would be good to add some supporting information). Cheers.2011-04-14
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    If you add a new definition to Wikipedia, please have a reference for it, preferably in the form of a published article.2018-12-17

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