(sum lnp * {lnm/lnp}) / (sum lnp) limit

Leroy Quet

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Dec 1, 2009, 12:46:12 PM12/1/09

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I am just wondering...

Let {x} = x - floor(x). Let lnx = natural logarithm of x.

Let A(m) = sum(p=primes <= m) lnp * {lnm/lnp}.
And B(m) = sum(p=primes <= m) lnp.

Then does the limit (as m approaches infinity) A(m)/B(m) exist?

If so, what is it?

0 <= A(m)/B(m) <= 1 for all m. But I don't know if the ratio converges
to a constant.

By the way, I know that A(m) = ln(m^pi(m) /LCM(1,2,3,...,m)), where pi
(m) is the number of primes <= m.

Thanks,
Leroy Quet

dan73

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Dec 1, 2009, 10:53:25 PM12/1/09

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Leroy,

What is your value for pi(25)?
Just so I can see what you are doing.

Dan

The Unbearable Likeness of Being

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Dec 2, 2009, 1:16:58 AM12/2/09

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Results 1 - 10 for What is your value for pi(25)?. (0.17 seconds)

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Leroy Quet

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Dec 2, 2009, 10:52:26 AM12/2/09

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The primes <= 25:
2,3,5,7,11,13,17,19,23. So, pi(25) = 9.
pi(29), on the other hand, would be 10.

pi(n) is common notation. I know it is confusing, but I didn't invent
it.

Thanks,
Leroy Quet

Leroy Quet

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Dec 2, 2009, 10:58:40 AM12/2/09

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Maybe the confusion comes from the fact that Google stuck a %#^$*!
carriage-return between "pi" and "(m)" in my original post.

Ignore that carriage return, of course.

Thanks,
Leroy Quet

rancid moth

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Dec 3, 2009, 12:09:29 AM12/3/09

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On Wed, 02 Dec 2009 07:58:40 -0800, Leroy Quet wrote:

> Maybe the confusion comes from the fact that Google stuck a %#^$*!
> carriage-return between "pi" and "(m)" in my original post.

[cut]

you need to get off and then stay away from google groups. download a copy
of PAN

http://pan.rebelbase.com/download/

(there are windows clients if you don't use linux)

and then get a free subscription to www.eternal-september.org