...

Riemann sphere:

imagine that you have a unit-diameter ball sitting on the origin of the Complex plane and you decide to

wrap that Complex plane around the ball in such a manner that all points on the graph that are a half-unit

away from the bottom point on the ball are mapped to the Equator of the that ball and all points at infinity

on the plane are mapped to the top-dead center of the unit ball and all the other points are uniquely distri-

buted around the shell of the ball; Reimann decided to define that that'd be a well-behaved mapping from

the Complex plane onto a unit ball.

...

lines of infinity:

now, imagine that we evaluate the function y= zeta(x)= 1/(1^x) +1/(2^x) +1/(3^x) + ... +1/(n^x) until 'n'

goes to infinity on the Real plane at the points x= 0 and x= 1; we would easily notice that two lines of

infinity appear on the Real plane that would have to translate over to the Complex plane, if it were super-

imposed onto the Real-valued graph. I don't know why zeta(0)= -1/2 when evaluated using the Complex

numbers... it shouldn't.

...

well-behaved mapping:

what Riemann imagined was this... if the zeta function were to be graphed on the Complex plane, then

the unit ball would need to be moved to the x= 1/2, y= 0 point such that both the Complex plane and

the Complex-valued zeta function would map to the unit ball in a well-behaved manner, only now the

unit ball has those two additional lines at infinity residing as left- and right-hand arcs on the unit ball

which join the left-hand pole and right-hand pole to top-dead center, separately. since all of the points

at infinity of the Complex plane map to the singularity atop the unit ball, then, additionally, the points

at infinity from the two arcs on the unit ball also have to 'stereographically' project onto the 'non-trivial'

zeros on a longitudinal Equator about the unit ball that are equally distant from those arcs at infinity.

Riemann is saying that the mapping must be bi-directional for all points at infinity.

...

he gracefully summarized this by saying...

...

"All 'non-trivial' zeros of the zeta function must have a real part equal to 1/2."

...

the geometry is not hard to visualize, but finding the inverse mappings would be... it would be analogous

to discussing Reiki w/a Japanese monk; "since you have hands, you must have a soul, and vice versa." we

shouldn't be questioning whether or not the zeros of the Complex-valued zeta function lie on the 1/2-line,

but rather we should try to describe the inverse 'stereographic' projective mapping of the arcs at infinity on

the Riemann sphere and its longitudinal Equator.

...

there's a connection... or the notion of a Riemann sphere would cease to exist; also the same process could

be used to describe the position of the unit-diameter ball on the Complex plane while considering other ra-

tional Complex functions, not just the zeta function.

...

here's a "realizeable" description of the Riemann Hypothesis:

...

if we take the zeta function... f(s)= summation[1/(n^s)], not worring about 'n' for the moment, and taking just the

1st derivative respective to 's', we would have (d/ds)(f(s))= (d/ds)sum[1/(n^s)]= sum[(d/ds)(n^(-s))]= sum[((-s)*

(n^(-s-1)))/(ln(-s))], and if the real part equals 1/g, we have that (d/ds)(f(s))== 0 everywhere; we'd be unable to

detect the slope of any solutions not appearing on the x-axis itself; there existance would be 'masked' by the

type of function that zeta is... and if the 2nd derivative and all subsequent derivatives cannot describe the con-

cavity, etc. of any solutions not appearing on the x-axis, then geometrically, we'd have a problem... those points

not on the x-axis become special (or non-trivial) in the sense that they must be mutually orthogonal to the x-axis

and to a set of points at infinity w/respect to the function.

...

The only way that can happen is... if these special points lie between the two lines that exhibit the points of in-

finity for the zeta function at x= 0 and x= 1. And, when these special points are placed onto Riemann's sphere,

they must be on the x=1/2 line, or longitudinal Equator to remain mutually orthogonal to the lines at x=0 and x=1,

also plotted on the Riemann sphere. It doesn't explain the connection between the zeta function and prime num-

bers, Euler did that; the geometry can be reasonably deduced by calculating the first few derivatives w/repsect

to 's'. Riemann must have referred to the geometry "only" and how this case could "only" arise in the situation of a

function on the Complex plane that "only" took on Complex values.

...

again, these individual, 'non-trivial' points when mapped from the solutions must be perpendicular to both the

x-axis and the lines of infinity that appear on the graph of the zeta function at the same time; it's possible... even

though examining these derivatives do NOT prove the existance of such a solution set nor any connection be-

tween the zeta function and the set of prime numbers. The fact they do exist has been calculated into the trillions

of points... and they will all fall between the two lines of infinity; it's believeable. I've just explained to you how

to identify the "problem" points as a new variety of analysis on Complex plane.

...

assuming that the set of 'non-trivial' points form some type of continuous curve between the two lines of infinity,

we'll see that it's "very impossible" to assign an area under the curve, so that all of the points, logically, must fall

on a vertical line. the integral of f(s) wrt 's' or Int(f(s)) = Int[sum(1/(n^s))] between s=0 to s=1 is equivalent to sum

[Int(f(s))]= sum[(n^(-s)/ln(n)] using the rule of integration for b^x where 'b' is some base number and 'x' is the gen-

eric variable, so that we end up with the summation of [1/(n*ln(n)) -1/ln(n)] over n=1 to some N which is the equi-

valent to an infinite sum of terms with the very first term being INFINITY! thus, the mysterious, 'non-trivial', or the

"problem" points as I call them, must fall on a single, vertical line just like Reimann imagined them, or we'd have

to deal with a huge area that doesn't begin to present itself!

...

in medical terms, it's known as a false-negative result... the typical integration between the two lines of infinity

says that the area is healthy and without bound, but a simple search shows that the 'non-trivial' points are sick

and must fall on a single vertical line; Riemann must have known that this was the case! again, if some 'non-

trivial' points didn't lie on the x= 1/2 line, then they wouldn't be mutually orthogonal, if placed onto the Riemann

sphere; the angles from those points would be different relative to the lines at infinity as mapped to the Riemann

sphere!

...

formally, if we partially differentiate the zeta function w/respect to its imaginary part, since s = x + t*i, then we'd

end up with the summation [ln(n)/((n^i)*ln(i))] over the Natural numbers 'n'; we can, however, calculate ln(i) since

Euler told us that e^(pi*i) = -1. thus, multiplying both sides by -i^2, we'd have -i^2*e^(pi*i)= i^2 which equals...

e^(pi*i)= i^2; taking the natural log of both sides gives us (pi*i)/2 = ln(i) and the summation of the partial deri-

vative of the zeta function w/respect to 't' would be infinite; we just had to access Euler's formula and work a

little to stumble upon the calculation for ln(i). Reimann was totally right! He was saying that the slope of the zeta

function, if the solutions don't lie on the x-axis, that they'd extend 'only' vertically, since the slope is (infinity)*(i).

...

ironically, many history books mention that Riemann worked with mathematicians that were opposed to people

that tried to use rather lengthy calculations to arrive at an explanation for a function's behavior, but mathematicians

of recent times still chose to calculate trillions of solutions... (very bad). if you had realized that the analysis of the

zeta function exhibited 'non-trivial' results with regard to standard calculus, then you would certainly not try to

correct the situation by incorporating other functions into the analysis w/the zeta function attached; it would be

circular thinking...

...

...

you can't be afraid to experiment in math, and you certainly can't tell yourself that something

is too difficult to answer, or you probably won't work hard to find an answer... the *gamma*,

Euler-Mascheroni, and 'e' constants look crappy 'cause I'm not using LaTeX, (and I don't care.)

...

... n

... _________

... \

... \ 1

let Z = lim } -----

... n->infinty / k

... /________

... k= 1

...

let *gamma* = Z -ln(n) = a/b; Z -a/b = ln(n), and... e^(Z -a/b) = n, and the subraction of a finite

fraction such as a/b from Z doesn't change the size or type of the infinite Z, so ignore it. now,

n(unknown) = e^1 *e^1 * ... *e^1 = (irrational)*(irrational)* ... *(irrational), and both the minuend,

Z, and the subtahend, ln(n), must be irrational, since 'n' is irrational, not because "ln" is a special

type of function.

...

thus, by contradiction, *gamma* is irrational, since it can't be represented as a fraction.

*QED

10/14/2016

...

the same goes for e = (1 +1/n)^n; it's irrational.

...

...

... a [ 1 ]n [ a ](1/n) n +1 [ a ](1/n)

let ----- = [ 1 + ----- ] ; [------] = ------------ ; n * [------] = n +1;

... b [ n ] [ b ] n [ b ]

...

...

... 1 [ a ](1/n) [ 1 ] [ a ] [ a ](1/n)

take (d/dn) to get... ----- * n * [-----] * [- ----- ] * ln [-----] + [-----] = 1.

... n [ b ] [ n^2 ] [ b ] [ b ]

...

...

... [ a ] { [ ln(a/b) ] }n

... [-----] * { 1 - [--------------] } = 1, but only 1*1 = 1; and let 'n' go to infinity!

... [ b ] { [ n^2 ] }

...

so,... a/b = 1; since 1*1= 1; 1*[ 1 - 0/n^2]^n = 1; but 1 =/= e; by contradiction, 'e' is irrational!

*QED

01/08/2012

...

similarly,... let (a/b) = 2^e; log(2)[a/b] = e; let Z= log(2)[a/b], (d/dn) [n*Z^(1/n)] = (d/dn) [n +1];

n*(1/n)*Z^(1/n)*(-1/n^2) + Z^(1/n) = 1; Z^(1/n)*[ 1 - 1/(Z*n^2) ] = 1; Z*[ 1- 1/(Z*n^2) ]^n = 1; &

take limit as n goes to infinity...

...

(a/b)= 2; so, a= 2, b= 1; this leads us to a contradiction! 2 =/= 2^e. thus, 2^e is irrational.

*QED

01/09/2012

...

...

the argument is almost exactly the same for e^pi where pi = 3.14159265...

...

...

... [ a ] { [ ln(a/b) ] }n*pi

... [-----] * { 1 - [--------------] } = 1, but only 1*1 = 1; and let 'n' go to infinity!

... [ b ] { [ pi*n^2 ] }

...

...

so,... a/b = 1; since 1*1= 1; 1*[ 1 - 0/(pi*n^2)]^n*pi = 1; but 1 =/= e^pi; so, e^pi is irrational!

...

*QED

01/09/2012

...

Riemann sphere:

imagine that you have a unit-diameter ball sitting on the origin of the Complex plane and you decide to

wrap that Complex plane around the ball in such a manner that all points on the graph that are a half-unit

away from the bottom point on the ball are mapped to the Equator of the that ball and all points at infinity

on the plane are mapped to the top-dead center of the unit ball and all the other points are uniquely distri-

buted around the shell of the ball; Reimann decided to define that that'd be a well-behaved mapping from

the Complex plane onto a unit ball.

...

lines of infinity:

now, imagine that we evaluate the function y= zeta(x)= 1/(1^x) +1/(2^x) +1/(3^x) + ... +1/(n^x) until 'n'

goes to infinity on the Real plane at the points x= 0 and x= 1; we would easily notice that two lines of

infinity appear on the Real plane that would have to translate over to the Complex plane, if it were super-

imposed onto the Real-valued graph. I don't know why zeta(0)= -1/2 when evaluated using the Complex

numbers... it shouldn't.

...

well-behaved mapping:

what Riemann imagined was this... if the zeta function were to be graphed on the Complex plane, then

the unit ball would need to be moved to the x= 1/2, y= 0 point such that both the Complex plane and

the Complex-valued zeta function would map to the unit ball in a well-behaved manner, only now the

unit ball has those two additional lines at infinity residing as left- and right-hand arcs on the unit ball

which join the left-hand pole and right-hand pole to top-dead center, separately. since all of the points

at infinity of the Complex plane map to the singularity atop the unit ball, then, additionally, the points

at infinity from the two arcs on the unit ball also have to 'stereographically' project onto the 'non-trivial'

zeros on a longitudinal Equator about the unit ball that are equally distant from those arcs at infinity.

Riemann is saying that the mapping must be bi-directional for all points at infinity.

...

he gracefully summarized this by saying...

...

"All 'non-trivial' zeros of the zeta function must have a real part equal to 1/2."

...

the geometry is not hard to visualize, but finding the inverse mappings would be... it would be analogous

to discussing Reiki w/a Japanese monk; "since you have hands, you must have a soul, and vice versa." we

shouldn't be questioning whether or not the zeros of the Complex-valued zeta function lie on the 1/2-line,

but rather we should try to describe the inverse 'stereographic' projective mapping of the arcs at infinity on

the Riemann sphere and its longitudinal Equator.

...

there's a connection... or the notion of a Riemann sphere would cease to exist; also the same process could

be used to describe the position of the unit-diameter ball on the Complex plane while considering other ra-

tional Complex functions, not just the zeta function.

...

here's a "realizeable" description of the Riemann Hypothesis:

...

if we take the zeta function... f(s)= summation[1/(n^s)], not worring about 'n' for the moment, and taking just the

1st derivative respective to 's', we would have (d/ds)(f(s))= (d/ds)sum[1/(n^s)]= sum[(d/ds)(n^(-s))]= sum[((-s)*

(n^(-s-1)))/(ln(-s))], and if the real part equals 1/g, we have that (d/ds)(f(s))== 0 everywhere; we'd be unable to

detect the slope of any solutions not appearing on the x-axis itself; there existance would be 'masked' by the

type of function that zeta is... and if the 2nd derivative and all subsequent derivatives cannot describe the con-

cavity, etc. of any solutions not appearing on the x-axis, then geometrically, we'd have a problem... those points

not on the x-axis become special (or non-trivial) in the sense that they must be mutually orthogonal to the x-axis

and to a set of points at infinity w/respect to the function.

...

The only way that can happen is... if these special points lie between the two lines that exhibit the points of in-

finity for the zeta function at x= 0 and x= 1. And, when these special points are placed onto Riemann's sphere,

they must be on the x=1/2 line, or longitudinal Equator to remain mutually orthogonal to the lines at x=0 and x=1,

also plotted on the Riemann sphere. It doesn't explain the connection between the zeta function and prime num-

bers, Euler did that; the geometry can be reasonably deduced by calculating the first few derivatives w/repsect

to 's'. Riemann must have referred to the geometry "only" and how this case could "only" arise in the situation of a

function on the Complex plane that "only" took on Complex values.

...

again, these individual, 'non-trivial' points when mapped from the solutions must be perpendicular to both the

x-axis and the lines of infinity that appear on the graph of the zeta function at the same time; it's possible... even

though examining these derivatives do NOT prove the existance of such a solution set nor any connection be-

tween the zeta function and the set of prime numbers. The fact they do exist has been calculated into the trillions

of points... and they will all fall between the two lines of infinity; it's believeable. I've just explained to you how

to identify the "problem" points as a new variety of analysis on Complex plane.

...

assuming that the set of 'non-trivial' points form some type of continuous curve between the two lines of infinity,

we'll see that it's "very impossible" to assign an area under the curve, so that all of the points, logically, must fall

on a vertical line. the integral of f(s) wrt 's' or Int(f(s)) = Int[sum(1/(n^s))] between s=0 to s=1 is equivalent to sum

[Int(f(s))]= sum[(n^(-s)/ln(n)] using the rule of integration for b^x where 'b' is some base number and 'x' is the gen-

eric variable, so that we end up with the summation of [1/(n*ln(n)) -1/ln(n)] over n=1 to some N which is the equi-

valent to an infinite sum of terms with the very first term being INFINITY! thus, the mysterious, 'non-trivial', or the

"problem" points as I call them, must fall on a single, vertical line just like Reimann imagined them, or we'd have

to deal with a huge area that doesn't begin to present itself!

...

in medical terms, it's known as a false-negative result... the typical integration between the two lines of infinity

says that the area is healthy and without bound, but a simple search shows that the 'non-trivial' points are sick

and must fall on a single vertical line; Riemann must have known that this was the case! again, if some 'non-

trivial' points didn't lie on the x= 1/2 line, then they wouldn't be mutually orthogonal, if placed onto the Riemann

sphere; the angles from those points would be different relative to the lines at infinity as mapped to the Riemann

sphere!

...

formally, if we partially differentiate the zeta function w/respect to its imaginary part, since s = x + t*i, then we'd

end up with the summation [ln(n)/((n^i)*ln(i))] over the Natural numbers 'n'; we can, however, calculate ln(i) since

Euler told us that e^(pi*i) = -1. thus, multiplying both sides by -i^2, we'd have -i^2*e^(pi*i)= i^2 which equals...

e^(pi*i)= i^2; taking the natural log of both sides gives us (pi*i)/2 = ln(i) and the summation of the partial deri-

vative of the zeta function w/respect to 't' would be infinite; we just had to access Euler's formula and work a

little to stumble upon the calculation for ln(i). Reimann was totally right! He was saying that the slope of the zeta

function, if the solutions don't lie on the x-axis, that they'd extend 'only' vertically, since the slope is (infinity)*(i).

...

ironically, many history books mention that Riemann worked with mathematicians that were opposed to people

that tried to use rather lengthy calculations to arrive at an explanation for a function's behavior, but mathematicians

of recent times still chose to calculate trillions of solutions... (very bad). if you had realized that the analysis of the

zeta function exhibited 'non-trivial' results with regard to standard calculus, then you would certainly not try to

correct the situation by incorporating other functions into the analysis w/the zeta function attached; it would be

circular thinking...

...

...

you can't be afraid to experiment in math, and you certainly can't tell yourself that something

is too difficult to answer, or you probably won't work hard to find an answer... the *gamma*,

Euler-Mascheroni, and 'e' constants look crappy 'cause I'm not using LaTeX, (and I don't care.)

...

... n

... _________

... \

... \ 1

let Z = lim } -----

... n->infinty / k

... /________

... k= 1

...

let *gamma* = Z -ln(n) = a/b; Z -a/b = ln(n), and... e^(Z -a/b) = n, and the subraction of a finite

fraction such as a/b from Z doesn't change the size or type of the infinite Z, so ignore it. now,

n(unknown) = e^1 *e^1 * ... *e^1 = (irrational)*(irrational)* ... *(irrational), and both the minuend,

Z, and the subtahend, ln(n), must be irrational, since 'n' is irrational, not because "ln" is a special

type of function.

...

thus, by contradiction, *gamma* is irrational, since it can't be represented as a fraction.

*QED

10/14/2016

...

the same goes for e = (1 +1/n)^n; it's irrational.

...

...

... a [ 1 ]n [ a ](1/n) n +1 [ a ](1/n)

let ----- = [ 1 + ----- ] ; [------] = ------------ ; n * [------] = n +1;

... b [ n ] [ b ] n [ b ]

...

...

... 1 [ a ](1/n) [ 1 ] [ a ] [ a ](1/n)

take (d/dn) to get... ----- * n * [-----] * [- ----- ] * ln [-----] + [-----] = 1.

... n [ b ] [ n^2 ] [ b ] [ b ]

...

...

... [ a ] { [ ln(a/b) ] }n

... [-----] * { 1 - [--------------] } = 1, but only 1*1 = 1; and let 'n' go to infinity!

... [ b ] { [ n^2 ] }

...

so,... a/b = 1; since 1*1= 1; 1*[ 1 - 0/n^2]^n = 1; but 1 =/= e; by contradiction, 'e' is irrational!

*QED

01/08/2012

...

similarly,... let (a/b) = 2^e; log(2)[a/b] = e; let Z= log(2)[a/b], (d/dn) [n*Z^(1/n)] = (d/dn) [n +1];

n*(1/n)*Z^(1/n)*(-1/n^2) + Z^(1/n) = 1; Z^(1/n)*[ 1 - 1/(Z*n^2) ] = 1; Z*[ 1- 1/(Z*n^2) ]^n = 1; &

take limit as n goes to infinity...

...

(a/b)= 2; so, a= 2, b= 1; this leads us to a contradiction! 2 =/= 2^e. thus, 2^e is irrational.

*QED

01/09/2012

...

...

the argument is almost exactly the same for e^pi where pi = 3.14159265...

...

...

... [ a ] { [ ln(a/b) ] }n*pi

... [-----] * { 1 - [--------------] } = 1, but only 1*1 = 1; and let 'n' go to infinity!

... [ b ] { [ pi*n^2 ] }

...

...

so,... a/b = 1; since 1*1= 1; 1*[ 1 - 0/(pi*n^2)]^n*pi = 1; but 1 =/= e^pi; so, e^pi is irrational!

...

*QED

01/09/2012

...