Fourier Analysis is the Cell theory of True Calculus #43 Uni-text 8th

which series, the Bernoulli or Fourier? #74 Math-Professor-text 8th ed.: TRUE CALCULUS

Sticking to the question of uniqueness. Which formula do we use? Do we use the Bernoulli formula which was discovered to solve the physics of vibrating string, Bernoulli formula:

y = a_1 sin (pi*x/L) cos (pi*c*t/L) + a_2 sin (pi*x/L) cos (pi*c*t/L) + . . .

Or, do we use the Fourier formula which was discovered to solve the physics of heat flow:

y = (1/2)*a_0 + Summation (a_n) *cos (n*x) + Summation (b_n) *sin (n*x)+ .. . .

Which ever, what we want is to convert all functions into just functions whose components are just simply sine and cosine. And I am calling both formulas as the Fourier theory in the idea that either formula captures all the functions of a Grid System and converts them into sine and cosine components.

Now here we can provide an analogy for the Grid Systems. So we have the 10 Grid and we have all 4 quadrants and we have the x and y axis go to only 10 for that is the border between finite and infinity in 10 Grid. Now we pretend the x-axis is the vibrating string and is able to be extended like a music string to reach the y=10 and let go and vibrate, or, to be plucked for y equals any finite number along the y axis. So that the string when plucked remains inside the 10 Grid and is a function described by that Bernoulli formula.

Now to me, the Bernoulli formula versus the Fourier formula reminds me of the tussle between the Riemann encoding and the Euler encoding in the Riemann Hypothesis. Here is an old post that brings forth that fight and tussle between the Riemann zeta and Euler zeta.

--- quoting old post of mine ---
Newsgroups: sci.math, sci.physics, sci.logic
From: Archimedes Plutonium
Date: Tue, 10 May 2011 22:19:32 -0700 (PDT)
Local: Wed, May 11 2011 12:19Â*am
Subject: Euler zeta Chapt2 Computer data on pseudosphere and 10^603 as infinity #471 Correcting Math 3rd ed
Reply | Reply to author | Forward | Print | Individual message | Show original | Remove | Report this message | Find messages by this author
On May 10, 5:05Â*pm, Archimedes Plutonium

- Show quoted text -
The above reminds me alot of what I had done in April 2011,
From: Archimedes Plutonium 
Date: Tue, 5 Apr 2011 11:22:13 -0700 (PDT) 
Local: Tues, Apr 5 2011 1:22Â*pm 
Subject: Euler zeta at quartic exp; Riemann Hypothesis is not a 
math conjecture; Correcting Math 3rd ed
On Apr 5, 1:04Â*am, Archimedes Plutonium
- Hide quoted text -
wrote:
On Apr 4, 1:09Â*am, Archimedes wrote:
(snipped to save space)
=



1.082323191269685768273684748931191264869947699510 2564184784009090648683447
309\

 57590858261697449300912209644732005074539147200373 592
=



1.082323226971998635622207794878671825605538376453 5625153845608734776218891
824


05987550590040448707759100161180552631547853212636 1381204565456019762008885
405


79503366377362000985335657768895339833765225165090 6951421869663804225781187
265

 683682147691800078
I had the Computer go out to the prime 199 to see if I could bump up


 that convergency to

 be 1.0824 rather than the reputed convergency of 1.0823.

Alright, the computer weighed in again by delivering what the first 
46 
terms of the Riemann zeta is 
in exp4 to compare with the above which is the 46th term of the Euler 
zeta. 
= {1, 1/16, 1/81, 1/256, 1/625, 1/1296, 1/2401, 1/4096, 1/6561, 
1/10000, 1/14641, 1/20736, 1/28561, 1/38416, 1/50625, 1/65536, 
1/83521, 1/104976, 1/130321, 1/160000, 1/194481, 1/234256, 1/279841, 
1/331776, 1/390625, 1/456976, 1/531441, 1/614656, 1/707281, 1/810000, 
1/923521, 1/1048576, 1/1185921, 1/1336336, 1/1500625, 1/1679616, 
1/1874161, 1/2085136, 1/2313441, 1/2560000, 1/2825761, 1/3111696, 
1/3418801, 1/3748096, 1/4100625, 1/4477456} 
= 
71603846070161235984974416572048882124033931427 408513209760256918517060039/ 
66157745782864854606586051262785807972331951694 504381871645179494400000000
= 
1.082319919199951718467614815110375956994451304 0893810814806532632698059273

Apparently the Euler zeta is larger than the Riemann zeta. Which is 
strange to me for I would have guessed 
the Riemann zeta would be for the longest time ahead of the Euler 
zeta.
But let us ask some questions about Riemann zeta, Euler zeta and 
sphere, pseudosphere area. So we see the 
pseudosphere area always appearing to lag behind the value of the 
sphere area. And it is my claim that at 
10^603 where pi is in stagnation of three zero digits in a row, that 
the pseudosphere area catches up and 
overtakes the sphere area for the first time.
So has anyone studied or payed attention to the Euler zeta versus 
Riemann zeta as to the question of when one 
is larger than the other term by term? It is easy to place the sphere 
and pseudosphere into 
B matrices theory because the length of the arms of pseudosphere is 
where we measure the area in 10, then 10^2 then 10^3 etc etc.. However 
for the zetas, B matrices can not be applied with meaning.
But there remains many questions such as whether they shift back and 
forth as one zeta becomes larger than the other zeta or whether the 
Euler zeta stays in the lead and remains there until a long ways out? 
Now since pi is related to the primes and since pi has three zero 
digits in a row for the first time at 10^603, one has to wonder if 
something special happens with the Euler zeta and Riemann zeta at 
10^603?
--- end of old post of mine ---

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

Now, the only decent search for AP posts on Google Newsgroups, is a search for 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

how does the three 0's in a row in pi at 10^603 affect the Euler zeta? #75 Math-Professor-text 8th ed.: TRUE CALCULUS

Alright the below is a repeat of #74 post which I made a few minutes ago. The question for professors of mathematics is an immediate question from the previous post. The question is, we easily see how the three zeroes in a row in pi allow for the pseudosphere area to catch up and briefly overtake the sphere area where no area contributions are given to the sphere when pi has zero digit. The question then is, since the Euler zeta of multiplication of primes stays always ahead of the Riemann zeta term by term, yet at Floor pi*10^603 where pi has 3 zero digits in a row, how does that 3 zero digits in a row boost the term value of the Riemann zeta so that it crosses over the value of the Euler zeta for the first time. In other words, how does 3 zero digits in a row in pi influence the density of primes in the 10^603 region?

Sticking to the question of uniqueness. Which formula do we use? Do we use the Bernoulli formula which was discovered to solve the physics of vibrating string, Bernoulli formula:

y = a_1 sin (pi*x/L) cos (pi*c*t/L) + a_2 sin (pi*x/L) cos (pi*c*t/L) + . . .

Or, do we use the Fourier formula which was discovered to solve the physics of heat flow:

y = (1/2)*a_0 + Summation (a_n) *cos (n*x) + Summation (b_n) *sin (n*x)+ .. . .

Which ever, what we want is to convert all functions into just functions whose components are just simply sine and cosine.

Now here we can provide an analogy for the Grid Systems. So we have the 10 Grid and we have all 4 quadrants and we have the x and y axis go to only 10 for that is the border between finite and infinity in 10 Grid. Now we pretend the x-axis is the vibrating string and is able to be extended like a music string to reach the y=10 and let go and vibrate, or, to be plucked for y equals any finite number along the y axis. So that the string when plucked remains inside the 10 Grid and is a function described by that Bernoulli formula.

Now to me, the Bernoulli formula versus the Fourier formula reminds me of the tussle between the Riemann encoding and the Euler encoding in the Riemann Hypothesis. Here is an old post that brings forth that fight and tussle between the Riemann zeta and Euler zeta.

--- quoting old post of mine ---
Newsgroups: sci.math, sci.physics, sci.logic
From: Archimedes Plutonium
Date: Tue, 10 May 2011 22:19:32 -0700 (PDT)
Local: Wed, May 11 2011 12:19Â*am
Subject: Euler zeta Chapt2 Computer data on pseudosphere and 10^603 as infinity #471 Correcting Math 3rd ed
Reply | Reply to author | Forward | Print | Individual message | Show original | Remove | Report this message | Find messages by this author
On May 10, 5:05Â*pm, Archimedes Plutonium

- Show quoted text -
The above reminds me alot of what I had done in April 2011,
From: Archimedes Plutonium 
Date: Tue, 5 Apr 2011 11:22:13 -0700 (PDT) 
Local: Tues, Apr 5 2011 1:22Â*pm 
Subject: Euler zeta at quartic exp; Riemann Hypothesis is not a 
math conjecture; Correcting Math 3rd ed
On Apr 5, 1:04Â*am, Archimedes Plutonium
- Hide quoted text -
wrote:
On Apr 4, 1:09Â*am, Archimedes wrote:
(snipped to save space)
=



1.082323191269685768273684748931191264869947699510 2564184784009090648683447
309\

 57590858261697449300912209644732005074539147200373 592
=



1.082323226971998635622207794878671825605538376453 5625153845608734776218891
824


05987550590040448707759100161180552631547853212636 1381204565456019762008885
405


79503366377362000985335657768895339833765225165090 6951421869663804225781187
265

 683682147691800078
I had the Computer go out to the prime 199 to see if I could bump up


 that convergency to

 be 1.0824 rather than the reputed convergency of 1.0823.
Alright, the computer weighed in again by delivering what the first 
46 
terms of the Riemann zeta is 
in exp4 to compare with the above which is the 46th term of the Euler 
zeta. 
= {1, 1/16, 1/81, 1/256, 1/625, 1/1296, 1/2401, 1/4096, 1/6561, 
1/10000, 1/14641, 1/20736, 1/28561, 1/38416, 1/50625, 1/65536, 
1/83521, 1/104976, 1/130321, 1/160000, 1/194481, 1/234256, 1/279841, 
1/331776, 1/390625, 1/456976, 1/531441, 1/614656, 1/707281, 1/810000, 
1/923521, 1/1048576, 1/1185921, 1/1336336, 1/1500625, 1/1679616, 
1/1874161, 1/2085136, 1/2313441, 1/2560000, 1/2825761, 1/3111696, 
1/3418801, 1/3748096, 1/4100625, 1/4477456} 
= 
71603846070161235984974416572048882124033931427 408513209760256918517060039/ 
66157745782864854606586051262785807972331951694 504381871645179494400000000
= 
1.082319919199951718467614815110375956994451304 0893810814806532632698059273
Apparently the Euler zeta is larger than the Riemann zeta. Which is 
strange to me for I would have guessed 
the Riemann zeta would be for the longest time ahead of the Euler 
zeta.
But let us ask some questions about Riemann zeta, Euler zeta and 
sphere, pseudosphere area. So we see the 
pseudosphere area always appearing to lag behind the value of the 
sphere area. And it is my claim that at 
10^603 where pi is in stagnation of three zero digits in a row, that 
the pseudosphere area catches up and 
overtakes the sphere area for the first time.
So has anyone studied or payed attention to the Euler zeta versus 
Riemann zeta as to the question of when one 
is larger than the other term by term? It is easy to place the sphere 
and pseudosphere into 
B matrices theory because the length of the arms of pseudosphere is 
where we measure the area in 10, then 10^2 then 10^3 etc etc.. However 
for the zetas, B matrices can not be applied with meaning.
But there remains many questions such as whether they shift back and 
forth as one zeta becomes larger than the other zeta or whether the 
Euler zeta stays in the lead and remains there until a long ways out? 
Now since pi is related to the primes and since pi has three zero 
digits in a row for the first time at 10^603, one has to wonder if 
something special happens with the Euler zeta and Riemann zeta at 
10^603?
--- end of old post of mine ---

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

Now, the only decent search for AP posts on Google Newsgroups, is a search for 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

On Wednesday, November 20, 2013 2:29:16 PM UTC-6, wrote:
how does the three 0's in a row in pi at 10^603 affect the Euler zeta? #75 Math-Professor-text 8th ed.: TRUE CALCULUS



Alright the below is a repeat of #74 post which I made a few minutes ago. The question for professors of mathematics is an immediate question from the previous post. The question is, we easily see how the three zeroes in a row in pi allow for the pseudosphere area to catch up and briefly overtake the sphere area where no area contributions are given to the sphere when pi has zero digit. The question then is, since the Euler zeta of multiplication of primes stays always ahead of the Riemann zeta term by term, yet at Floor pi*10^603 where pi has 3 zero digits in a row, how does that 3 zero digits in a row boost the term value of the Riemann zeta so that it crosses over the value of the Euler zeta for the first time. In other words, how does 3 zero digits in a row in pi influence the density of primes in the 10^603 region?



Sticking to the question of uniqueness. Which formula do we use? Do we use the Bernoulli formula which was discovered to solve the physics of vibrating string, Bernoulli formula:



y = a_1 sin (pi*x/L) cos (pi*c*t/L) + a_2 sin (pi*x/L) cos (pi*c*t/L) + . . .



Or, do we use the Fourier formula which was discovered to solve the physics of heat flow:



y = (1/2)*a_0 + Summation (a_n) *cos (n*x) + Summation (b_n) *sin (n*x)+ . . .



Which ever, what we want is to convert all functions into just functions whose components are just simply sine and cosine.



Now here we can provide an analogy for the Grid Systems. So we have the 10 Grid and we have all 4 quadrants and we have the x and y axis go to only 10 for that is the border between finite and infinity in 10 Grid. Now we pretend the x-axis is the vibrating string and is able to be extended like a music string to reach the y=10 and let go and vibrate, or, to be plucked for y equals any finite number along the y axis. So that the string when plucked remains inside the 10 Grid and is a function described by that Bernoulli formula.



Now to me, the Bernoulli formula versus the Fourier formula reminds me of the tussle between the Riemann encoding and the Euler encoding in the Riemann Hypothesis. Here is an old post that brings forth that fight and tussle between the Riemann zeta and Euler zeta.



--- quoting old post of mine ---

Newsgroups: sci.math, sci.physics, sci.logic

From: Archimedes Plutonium

Date: Tue, 10 May 2011 22:19:32 -0700 (PDT)

Local: Wed, May 11 2011 12:19Â*am

Subject: Euler zeta Chapt2 Computer data on pseudosphere and 10^603 as infinity #471 Correcting Math 3rd ed

Reply | Reply to author | Forward | Print | Individual message | Show original | Remove | Report this message | Find messages by this author

On May 10, 5:05Â*pm, Archimedes Plutonium



- Show quoted text -

The above reminds me alot of what I had done in April 2011,

From: Archimedes Plutonium 
Date: Tue, 5 Apr 2011 11:22:13 -0700 (PDT) 
Local: Tues, Apr 5 2011 1:22Â*pm 
Subject: Euler zeta at quartic exp; Riemann Hypothesis is not a 
math conjecture; Correcting Math 3rd ed

On Apr 5, 1:04Â*am, Archimedes Plutonium

- Hide quoted text -

wrote:

On Apr 4, 1:09Â*am, Archimedes wrote:

(snipped to save space)

=






1.082323191269685768273684748931191264869947699510 2564184784009090648683447

309\


 57590858261697449300912209644732005074539147200373 592

=






1.082323226971998635622207794878671825605538376453 5625153845608734776218891

824




05987550590040448707759100161180552631547853212636 1381204565456019762008885

405




79503366377362000985335657768895339833765225165090 6951421869663804225781187

265


 683682147691800078

I had the Computer go out to the prime 199 to see if I could bump up




 that convergency to


 be 1.0824 rather than the reputed convergency of 1.0823.

Alright, the computer weighed in again by delivering what the first 
46 
terms of the Riemann zeta is 
in exp4 to compare with the above which is the 46th term of the Euler 
zeta. 
= {1, 1/16, 1/81, 1/256, 1/625, 1/1296, 1/2401, 1/4096, 1/6561, 
1/10000, 1/14641, 1/20736, 1/28561, 1/38416, 1/50625, 1/65536, 
1/83521, 1/104976, 1/130321, 1/160000, 1/194481, 1/234256, 1/279841, 
1/331776, 1/390625, 1/456976, 1/531441, 1/614656, 1/707281, 1/810000, 
1/923521, 1/1048576, 1/1185921, 1/1336336, 1/1500625, 1/1679616, 
1/1874161, 1/2085136, 1/2313441, 1/2560000, 1/2825761, 1/3111696, 
1/3418801, 1/3748096, 1/4100625, 1/4477456} 
= 
71603846070161235984974416572048882124033931427 408513209760256918517060039/ 
66157745782864854606586051262785807972331951694 504381871645179494400000000

= 
1.082319919199951718467614815110375956994451304 0893810814806532632698059273

Apparently the Euler zeta is larger than the Riemann zeta. Which is 
strange to me for I would have guessed 
the Riemann zeta would be for the longest time ahead of the Euler 
zeta.

But let us ask some questions about Riemann zeta, Euler zeta and 
sphere, pseudosphere area. So we see the 
pseudosphere area always appearing to lag behind the value of the 
sphere area. And it is my claim that at 
10^603 where pi is in stagnation of three zero digits in a row, that 
the pseudosphere area catches up and 
overtakes the sphere area for the first time.

So has anyone studied or payed attention to the Euler zeta versus 
Riemann zeta as to the question of when one 
is larger than the other term by term? It is easy to place the sphere 
and pseudosphere into 
B matrices theory because the length of the arms of pseudosphere is 
where we measure the area in 10, then 10^2 then 10^3 etc etc. However 
for the zetas, B matrices can not be applied with meaning.

But there remains many questions such as whether they shift back and 
forth as one zeta becomes larger than the other zeta or whether the 
Euler zeta stays in the lead and remains there until a long ways out? 
Now since pi is related to the primes and since pi has three zero 
digits in a row for the first time at 10^603, one has to wonder if 
something special happens with the Euler zeta and Riemann zeta at 
10^603?

--- end of old post of mine ---





#75.1 how does the three 0's in a row in pi at 10^603 affect the Euler zeta? #75 Math-Professor-text 8th ed.: TRUE CALCULUS

Alright, it is clear to even High School students that the surface area of pseudosphere when always below the value of the associated sphere surface area would gain on the sphere when the digits of pi are 0. So that in pi at 10^-603 with its 3 zero digits in a row would no longer be adding on more surface area, but the pseudosphere whose distance in its two arms are increasing and thus adding on more surface area as those arms go out further. So it is intuitive how the pseudosphere area overtakes and crosses over the area of the associated sphere area.

It is not intuitive how the Riemann zeta can overtake and crossover the Euler zeta when pi has 3 zero digits in a row. But there is help from an old known series of pi as that of this:

pi = 4 - 4/3 + 4/5 - 4/7 + 4/11 - 4/13 + 4/17 - . . .

Now if you notice the series relates pi to primes, and in that relationship we can sort of surmise a density of primes as pi digits increase. So we ask the question, how many primes do we have to use before our pi is 3.1 and the answer I have from my calculator is that we have to go to the 4/17 term in order to reach pi = 3.1 for the first time. So this tells us a density of primes in the Euler zeta. And this forebodes what happens at pi in the 3 zero digits in a row at 10^-603. That the density of primes in that region are the lowest density of primes ever in pi up to that moment in pi.

So, where the 3 zero digits in a row in pi allowed the pseudosphere surface area to crossover that of the associated sphere area for the first time in pi, the 3 zero digits in a row allow the Riemann zeta to crossover the Euler zeta for the first time in pi. Which means of course, that this is the borderline of where finite ends and infinity starts.

Now I no longer have access to a computer which can calculate the exact crossover of the pseudosphere in the 10^-603 region, nor access to a computer that can show the density of primes being lowest density in pi at 10^-603, nor access to a computer that gives where exactly the Riemann zeta crosses over the value of the Euler zeta in that region.

But, many professors of math do have that access and can provide the exact answers I ask of above. Good luck!

AP

It is not intuitive how the Riemann zeta can overtake and crossover the Euler zeta when pi has 3 zero digits in a row. But there is help from an old known series of pi as that of this:



pi = 4 - 4/3 + 4/5 - 4/7 + 4/11 - 4/13 + 4/17 - . . .


Alright now, the above is a mistake by me on intention, for the real true formula is all odd numbers starting with 3 in the denominator, for I missed terms like 4/9 and 4/15 etc etc.

But the logic I am proposing, in that logic we can disregard the odd composite numbers.

All we want is a measure of density of primes alone and the above formula provides that measure of density of primes as we move along the digits of pi..

So, I did make a mistake in saying that the formula above is equal to pi for it is wrong, but I did not make a mistake in the logic of the measuring tool that the formula provides, because we simply ignore the odd composite numbers. My aim is to gather information of prime density in the region of where pi has those first three zero digits in a row. Now there maybe even easier methods and tricks to pull out a prime density in the 10^-603 region. The above is just one method.

AP

On Thursday, November 21, 2013 9:17:52 AM UTC-6, David Bernier wrote:
On 11/21/2013 03:29 AM, Archimedes Plutonium wrote:


(snipped)




Do you believe in limits?


Hi David, no I do not accept a limit concept when finite versus infinity is borderless. The limit is a patch on lazy and dumb mathematicians who want finite and who want infinity but never want a border concept to precisely define what it means to be finite and what it means to be infinite.

So, David, in your Calculus you have dy/dx where dx = 0 and division by zero, all allowed because you have a phony limit concept. You have integrals with summation of rectangles all of which have no width, have 0 width of rectangles of no interior area, all allowed because you have a phony patch of a limit concept.

The limit concept of Calculus is just as phony as the limit of series.

So no, I cannot buy phony math concepts.



zeta(s) = lim_{n- oo} sum_{k = 1 ... n} 1/k^s



for s with Re(s)1. *That's Euler's zeta as an infinite sum.



zeta(s) = lim_{n- oo} product_{k = 1 ... n} (1 - (p_k)^(-s))^(-1)





for s with Re(s)1, p_k being the k'th prime.



It is easiest to understand the Euler product identity

for s real, with s1.



The validity of the Euler product identity

(that both expressions are the same) is a consequence

of the fundamental theorem of arithmetic on

factorization of integers into primes.




(snipped)

Under David's phony limit concept, he would agree that series A = 1 + 2 + 3 + 4 + ..... is equal to series B = 2 + 4 + 6 + 8 + .... for the reason that both go to infinity where infinity is not given a precise definition by giving a borderline.

David, the reason I bring up the Riemann Hypothesis of its two zeta functions is precisely to find out that borderline. The pseudosphere and sphere can find that border to be in the 10^603 region due to the 3 zero digits in pi. The RH can do the same, probably, for I am not certain of it, but likely do the same for the borderline between finite and infinity as the pseudosphere and sphere did with pi. With the RH, it is no longer pi that is the entity, but rather it is "e" in the 10^603 region that causes the Crossover.

Now David, I do believe in Convergence and Old Math did a fine job on Convergence and I have no argument against convergence concept, I do view the limit as totally phony.

So, here is series C = 1 + 0 + 0 + 0 + . . . . and it converges to 1 at infinity. And here is series D = 1 * 1 * 1 * 1 * . . . And it converges to 1 at infinity. You would agree and you would agree that C = D.

In Old Math, they had convergence theory correct, but they had equality of series another messy and vague and obscure concept, along with a phony limit concept.

So, how do we define Equality between two Series? This is something that the famous Euler never even dared to do because he never could examine finite versus infinity with a borderline between them. Euler could have come up with 10^603 because the sphere and pseudosphere were known to him for the pseudosphere dates to 1639 with Huygens and Euler lived most of the 1700s. So that Euler could have reasoned one day in the 1700s and said to himself, "look, if the surface area of sphere and pseudosphere were equal at infinity, then the borderline between finite and infinity is where the area of the pseudosphere crosses over that of the sphere as we peel away the digits of pi one by one peel them away and find a crossover juncture." Euler could have done the same with his Zeta function and said, "look, if the addition zeta was equal to the multiplication zeta, then at some juncture of the terms in those two zetas, the addition zeta will make a crossover to the multiplication zeta."

But Euler was too busy for that, unfortunately, for if he had found the borderline of finite and infinity, he could have spared all the phony mathematics that followed after Euler.

So for exponent 2 in the Riemann zeta we have the first few terms as:

1 + 1/(2^2) + 1/(3^2) + 1/(4^2) + 1/(5^2) + ...

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) * ...

So, what Euler and Riemann said was that those two series were equal at infinity, but neither Euler or Riemann ever defined what it means to be finite versus infinite. So we cannot say that the RH is even a piece of mathematics when it is full of phony concepts and unrefined definitions.

We can say that the Riemann zeta equals the Euler zeta if provided there is a term by term analysis where the Riemann zeta which is always below the value of the Euler zeta, all of a sudden crosses over the value. At that juncture, like the sphere and pseudosphere, we can say the zetas are equal and we can say that is the border of finite and infinity.

So let us try our hand for a few term analysis.

If we go by strictly TERM-BY-TERM we see the Riemann zeta always lag behind the Euler zeta except for perhaps when it reaches the 10^603 region where the primes are more sparse and far larger.

Or an alternative is to consider the zetas not as term by term but rather as what ceiling number for the terms so that if the ceiling number is 10, then the Riemann zeta has 10 terms while the Euler zeta has only 4 primes from 1 to 10 and thus has only 4 terms to be accounted for.

_________
When the Ceiling Progression number in both zetas is the number 2 we have the Riemann zeta as this:

1 + 1/(2^2) which equals 1.25

And for Progression number 2 in the Euler zeta we have this:

1/(1-1/(2^2) which is equal to 4/3 which is far larger than 1.25

_________
When the Ceiling Progression number in both zetas is the number 3 we have the Riemann zeta as this:

1 + 1/(2^2) + 1/(3^2) which equals 5/4 + 1/9 = 45/36 + 4/36 = 49/36

And for Progression number 3 in the Euler zeta we have this:

1/(1-1/(2^2) * 1/(1-1/(3^2) which is equal to 4/3 * 9/8 = 36/24
_________
When the Ceiling Progression number in both zetas is the number 4 we have the Riemann zeta as this:

1 + 1/(2^2) + 1/(3^2) + 1/(4^2) which equals 5/4 + 1/9 + 1/16 = 49/36 + 1/16 = 784/576 + 36/576 = 820/576

And for Progression number 4 in the Euler zeta we have the same as in previous since 4 is not prime:

1/(1-1/(2^2) * 1/(1-1/(3^2) which is equal to 4/3 * 9/8 = 36/24

__________


So now, which is it? TERM by TERM or Ceiling Progression Numbers? Which if any of those has a crossover?

AP

Ceiling Progression of zeta terms works to find infinity border #75.6 Math-Professor-text 8th ed.: TRUE CALCULUS

I believe we can go by Ceiling Progression and not need the term-by-term progression:


If we go by strictly TERM-BY-TERM we see the Riemann zeta always lag behind the Euler zeta except for perhaps when it reaches the 10^603 region where the primes are more sparse and far larger.



Or an alternative is to consider the zetas not as term by term but rather as what ceiling number for the terms so that if the ceiling number is 10, then the Riemann zeta has 10 terms while the Euler zeta has only 4 primes from 1 to 10 and thus has only 4 terms to be accounted for.



_________

When the Ceiling Progression number in both zetas is the number 2 we have the Riemann zeta as this:



1 + 1/(2^2) which equals 1.25



And for Progression number 2 in the Euler zeta we have this:



1/(1-1/(2^2) which is equal to 4/3 which is far larger than 1.25



_________

When the Ceiling Progression number in both zetas is the number 3 we have the Riemann zeta as this:



1 + 1/(2^2) + 1/(3^2) which equals 5/4 + 1/9 = 45/36 + 4/36 = 49/36



And for Progression number 3 in the Euler zeta we have this:



1/(1-1/(2^2) * 1/(1-1/(3^2) which is equal to 4/3 * 9/8 = 36/24

_________

When the Ceiling Progression number in both zetas is the number 4 we have the Riemann zeta as this:



1 + 1/(2^2) + 1/(3^2) + 1/(4^2) which equals 5/4 + 1/9 + 1/16 = 49/36 + 1/16 = 784/576 + 36/576 = 820/576



And for Progression number 4 in the Euler zeta we have the same as in previous since 4 is not prime:



1/(1-1/(2^2) * 1/(1-1/(3^2) which is equal to 4/3 * 9/8 = 36/24



__________





So now, which is it? TERM by TERM or Ceiling Progression Numbers? Which if any of those has a crossover?




I did the math for Ceiling 10 where the Euler zeta has 4 terms since there are only 4 primes in 1 to 10, but where the Riemann zeta has 10 terms, yet the Riemann zeta lags far behind, and where 1 to 10 is the most dense pocket of primes in the progression of 10 then 100 then 10^3 on up to 10^603.

Here is my math:

For the Riemann zeta

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


So, what I surmise from the above is that Ceiling progress for 100 still turns up the Euler zeta far ahead of the Riemann zeta and in fact all exponents of 10 until we reach 10^603 does the amount of terms of the Riemann zeta become critical, so much so that the amount of terms causes a Cross-over event.

Now I suspect no computer can do that summation of the zetas for 10^603 is too large a number.

What I do expect is some analysis can guide us to showing that the Crossover exists in 10^603.

AP