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July 3rd, 2015, 12:01 AM   #1
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Help Me Understand Riemann's 1859 Paper

http://www.claymath.org/sites/default/files/ezeta.pdf

There are quite a few things that are left unstated.

Notably on page 2 (labeled page 1 in the bottom but it is page 2 in the PDF) for his first equation he does not bother putting "infinity" over his Sigma or an "n=1" below it and doesn't do so for the Product Series either. That one is not a problem since I understand what that means from being familiar with studying the proof of this relationship before.

However, the second equation (same page) is confusing. I am familiar with improper integrals. However, "n" was just used in a context where it means a value that starts at 1 and goes up by 1 again and again to infinity, and now it's in an integral. The integral does not have a "Sigma" inside of it. Is this just an unstated thing?

Also, the Product Series sign on the right side of the equation still does not have any indices to indicate the initial or final values. If it is still p=1 at the bottom then that would mean since it is BigPi(s-1) that it would be (s-1)^infinity. Is that what he means here? Or is the lower index supposed to be s=1 now? Also is the n^s at the bottom supposed to include an unstated Sigma before it?

Maybe once I know the semantics of the 2nd equation the relationship will be obvious to me, but I'm not sure. I would like to know where Riemann got it from. Since he prefaces the 2nd equation with "On making use of the equation", I can only surmise that this is an equation proven in a previous paper either by Riemann or by another mathematician.

Last edited by skipjack; July 3rd, 2015 at 09:52 AM.
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July 3rd, 2015, 06:17 AM   #2
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I think it is clear from the fact that this is "as one substitutes for p all prime numbers" that the product on the left is over all prime numbers, p, while the sum is for n going from 1 to infinity.
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July 3rd, 2015, 11:09 AM   #3
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Quote:
Originally Posted by Country Boy View Post
I think it is clear from the fact that this is "as one substitutes for p all prime numbers" that the product on the left is over all prime numbers, p, while the sum is for n going from 1 to infinity.
Thank you, but I already understand the first equation. What confuses me is the second equation, the one with the integral on the left and a Product series in the numerator on the right side over an "n^s" in the denominator.

The integral uses "n" in its integrand but the integral does not contain a "Sigma" in it in spite of the most previous use of "n" being in an equation where "n" represents the variable which the index is substituted for in the infinite series. That makes me wonder if the "n" here has a different meaning than the "n" in the first equation or if the Sigma is left unstated under the integral in the second equation.

Another confusing part is whether BigPi(s-1) on the right side of the second equation is uses "p" as the variable for the index [in which case the absence of p would make BigPi(s-1) equal to (s-1)^(infinity)].

And finally does the "n^s" in the denominator on the right side mean Riemann means "Sigma (1/n^s)" multiplied by "BigPi(s-1)"?

Is this a common practice in mathematics (or was at the time?) to omit "Sigma" in cases like this?

Also based on Riemann's language I suspect the second equation was derived from a previous paper, though Riemann leaves that unstated. If anyone knows where I can find the second equation in its original paper please let me know.
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