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Topic: simple proof that the Riemann zeta never equals the Euler zeta #77
Math-Professor-text 8th ed.: TRUE CALCULUS

Replies: 4   Last Post: Dec 6, 2013 1:15 PM

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 plutonium.archimedes@gmail.com Posts: 18,572 Registered: 3/31/08
simple proof that the Riemann zeta never equals the Euler zeta #77
Math-Professor-text 8th ed.: TRUE CALCULUS

Posted: Dec 5, 2013 3:07 PM

Alright, I leave this with the math professor community to tussle over. Of course, the error of Euler ever thinking the two zetas are equal is because Euler never bothered to find a finite to infinity borderline and thus the algebra involved in placing series terms on one side of the equation and leaving other terms on the other side of the equation are invalid, for example the two series 1+ 3 + 5 + 7 + ..... and 0 + 2 + 4 + 6 +....
without a precise infinity definition would appear to be equal when "going to infinity" and once we say they are equal and do algebra manipulations so that we extract a multiplication series out of those two we can come to the same false conclusion that Euler came to with his zeta functions.

Some years back, I pinpointed the flaw of Euler in his deriving of the equality of the zetas, for the flaw is that they are not equal and Euler made a mistake of algebra.

But here, let me provide a proof that the two zetas cannot ever be equal.

I am going to use the 10 Grid where we pretend 10 is the finite borderline and beyond are all infinity-numbers.

For the Riemann zeta with exponent 2.

1 + 1/(2^2) + 1/(3^2) + 1/(4^2) + 1/(5^2) + 1/(6^2) + 1/(7^2) + 1/(8^2) + 1/(9^2) + 1/(10^2)

in decimal form it is

1 + .25 + .111 + .062 + .04 + .027 + .020 + .015 + .012 + .01 for a grand total of approximately 1.547

And for the Euler zeta we have:

1/(1-1/(2^2) * 1/(1-1/(3^2) * 1/(1-1/(5^2) * 1/(1-1/(7^2)

1.333 * 1.125 * 1.041 * 1.020 = approx 1.592

Now, in New Math or Old Math, we can define whether a fraction is even or odd and use that even versus odd laws of mathematics.

Notice that all the terms of the Euler zeta for exponent 2 are odd numbers and when you multiply odd by odd your answer is always odd.

Now it is purported that the convergence of the above is pi^2/6.

Now pi itself is even for it is evenly divisible by 120 and so should pi^2/6
be evenly divisible at infinity, yet we see that no term of Euler zeta is ever going to be even due to the primes involved and all of those terms are odd.
So it cannot be true that the Euler zeta equals the Riemann zeta.

QED

--
Drexel's Math Forum has done an excellent search engine for author posts as seen here:
http://mathforum.org/kb/profile.jspa?userID=499986

Now, the only decent search for AP posts on Google Newsgroups, is a search for plutonium.archimedes@gmail.com for it brings up posts that are mostly authored by me and it brings up only about 250 posts. Whereas Drexel brings up nearly 8,000 AP posts. Old Google under Advanced Search
for author, could bring up 20,000 of my authored posts but Google is deteriorating in quality of its searches, likely because AP likes an author search and Google does not want to appear as satisfying to anything that AP likes. If AP likes something, Google is quick to change or alter it.

So the only search engine today doing author searches is Drexel. Spacebanter is starting to do author archive lists. But Google is going in the opposite direction of making author archived posts almost impossible to retrieve.

All the other types of Google searches of AP are just top heavy in hate-spam posts due to search-engine-bombing practices by thousands of hatemongers who have nothing constructive to do in their lives but attack other people.

Now one person claims that Google's deteriorating quality in searches of science newsgroups is all due to "indexing". Well, that is a silly excuse in my opinion, because there is no indexing involved when one simply asks for a author search. No indexing involved if one wants only the pure raw complete list of all posts by a single author. And Google is called the best search engine of our times, yet I have to go to Drexel to see 8,000 of my posts of which I had posted 22,000 to 36,000 posts from 1993 to 2013. It is a shame that Drexel can display 8,000 while Google has a difficult time of displaying 250 of my authored posts. Where the premiere search engine of Google is outclassed by Drexel and even by Spacebanter.

Archimedes Plutonium