It is well-known in high-level mathematics that:

(Euler-Mascheroni Constant)

(Meissel-Mertens Constant)

Is there a divergent series such that:

If so, can it be generalized to ?

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- Oct 18th 2009, 01:48 PMredsoxfan325Divergence ~ ln(ln(ln x))
It is well-known in high-level mathematics that:

(Euler-Mascheroni Constant)

(Meissel-Mertens Constant)

Is there a divergent series such that:

If so, can it be generalized to ? - Oct 19th 2009, 01:51 AMOpalg
The approximation can be justified by applying the procedure of the integral test for convergence to get .

A similar procedure will give you , , , and so on.

**Edit.**See corrections below. - Oct 19th 2009, 02:26 AMchisigma
Is...

(1)

... all right!...

Is...

(2)

... all right!...

But is...

(3)

... something wrong from me?(Thinking) ...

Kind regards

- Oct 19th 2009, 03:17 AMOpalg
- Oct 19th 2009, 10:43 AMredsoxfan325
Is there a meaningful infinite set , such that

diverges ~

Like maybe let

But, for all I know, that could converge. (I know that if , then the series does converge.)

Actually, now that I think about it, that sum must converge, because

EDIT: Although, the above calculation seems to imply that if we let then diverges. - Oct 21st 2009, 09:58 PMchiph588@
diverges because one can show that given and prime , .

- Oct 21st 2009, 10:01 PMredsoxfan325
What's ? (And also, where does my proof of convergence go wrong?)

- Oct 21st 2009, 10:06 PMchiph588@
- Oct 21st 2009, 10:11 PMredsoxfan325
- Oct 21st 2009, 10:25 PMchiph588@
I'd say this equality is incorrect:

. This is because for a fixed (or ), the latter sum doesn't take into account how many times the (or ) appear in forming an arbitrary .