Proof of Reimann Hypothesis Geometrical and others in pdf

P

roof

for

Riemann

H y p o t h e s i s.

( Mathew   Mathew Cherian BE, MBA(Western Michigan University.) on private independent research.)

hypothesis , Abstract: Proof of Riemann’s, hypothesis

that the real real part of the solution of ξ function is ½ is proved. Secondly Secondly all the roots of the Riemann ξ functio n lie own the ½ plane is also

 proved. Historical development of this area of Mathematics from Gauss, Legrange, Euler,

 Riemann to Hilbert is discussed. discussed. Initially. a surrogate surrogate differential equation equation for ξ function is derived and taking the roots of the differential equation, it is proved that the real trivial part 

of Riemann ξ function is ½. ½ . Also Riemann’s ξ =∫ 1/Ln(x) between limits 2 and ‘n’, gives the number of Prime numbers between 0 and that number x. So I choose the original ξ function of   Riemann to prove it. This is given in the book “God created Integers” by Prof. Stephen  Hawkins published by Penguin which is a good reference for this. A second part of the proof  requires that the non-trivial complex part of the root must lie on the ½ plane which is also

 proved by proving the Elliptical nature of the ξ function. f unction. Please refer any books on basic

building blocks of, conic sections and their relationships and roots for Parabolic, Hyperbolic and Elliptical functions. There is a good book by Prof. Hardy on, this, on’ Complex  Numbers’. Two differential equations are derived which are similar to each other with different constant  terms ‘c’. One is used to calculate the constant term of the other. This method is employed to widely used in Engineering. The elicit the elliptical nature of Rieman n ξ function which is widely application of which is given in two different papers of mine separately.

Part I.

Legrange and Gauss conjured that п(x) the function counting all the primes less than x

asymptotically approaches Li(x) n Li(x) = ∫ 2

dx/ln(x)

meaning п(x)/Li(x) tend to 1, where,

Euler created a time series solution to the function Li(x) and Riemann named it the ξ function adding his own solution to Euler’s work.

In Riemann’s words “a value x is the root of a function f(x) if f(x)=0. A root of the function

ξ(x) is real if and a nd only if the root of the zeta function is complex number with real part equal to ½”.

Proving the real part to be ½ was left undone by Riemann. Riemann. Hilbert later on added, finding finding the proof for Riemann hypothesis as one of the problem that remain un resolved in Mathematics. (Ref.Stephen Hawkins)

Also Riemann’s Hypothesis conjures that all the non- trivial roots of Riemann’s ξ function lie on the ½ plane, which also need be proved according to Prof. Calvin Clawson(Ref).

So the conjunctures proved are 1)the trivial root of Riemann ξ function is at ½ and a nd 2) the non-trivial roots are all on the t he ½ plane. Riemann Zeta function’s Formation from Gauss’s conjuncture. Consider a Sine wave.

√2 Sine 45 = 1/ √2 0

Cosec 45o=√2 = (1/ln 2 ), which gives the primes pr imes at 2.

approximately gives the primes pr imes before 2 which is like Now, ∫1/ln(2)=1.414+1+1 = 3 approximately integrating 1/ln(x) or ∫1/ln(x) Continuing like this if we substitute x by 3,4 ….. etc; we see,

∫1/ln(x) gives the primes before x, which is what Riemann ξ function first part is all about. For Eg; ∫1/ln(3)=1+1+0.686+1.1=4

Heuristic Proof  Consider Consider a Cycloid, C ycloid, transiting in Sine formation. The, time t ime period for it is ∏ radians. The peak of it is seen at

∏/2, where where we find find the non-trivial root. Now, if we remove remove the common common

factor ∏ from the time scale to,

convert the time scale to numerical from trigonometric, we

see that the trivial root is at numerical 1/2.

Note: We can say from the t he above that the Lactus Rectum (inside the ellipse at ½ where t he roots appear) of the Ellipse is at ½ which is the constant term ‘c’ of the differential equation elicited later below. ‘a’ and ‘b’ constants co nstants of the differential equation are the x-axis eccentricity eccentricity and the t he y-axis of the ellipse.

Part 2. Proof 

To get an initial peep into Riemann ξ function, we consider the entire number line as consisting of concentric circles with the number as half t he diameter of twice the number. That is the number become the radius. Draw an isosless triangle above the diameter. No w construct an equilateral triangle inside the isosless triangle. Now take the number 4 and radius becomes 2. The corner of the equilateral triangle is seen to be close to the Natural Logrithm of 2 with approximately an error of 25%. So we can write, Ln(x)= (1.25(x(2-√2))/2 = 0.36612.x As we go up the number circles we see that the logarithm may not be (2- √2) distance fr om om the corner of the t he isosless isosless triangle drawn, but this overcome by figuring out t he appropriate appropriate growth rate down below called ‘a’. Reciprocating LHS and RHS and integrat ing,

∫1/Ln(x)=∫1/0.36612.x =1/0.36612∫1/x =ln(x)/0.36612 + (x/11)e0.5 where the second term is the error t erm and so the whole RHS can be written as, Ln(x)/0.36612+(x/11)e0.5= a.(ln(x)/0.36612) Here ’ a’ is found to be, 0.5

a= 1+(x.e .0.36612/(ln(x).11)) Therefore Riemann ξ function, funct ion,

∫1/Ln(x)= ln(x)/0.36612[1+x/ln(x)(0.054875), simplifying simplifying we get, get , =(ln(x)/0.36612)+(x/0.36612)(0.54875) =(ln(x)/0.36612)+(x(0.14988)) ie; ∫1/ln(x).dx=(ln(x)/0.36612)+0.14988.x

This shows that there is a ‘conservation law’ for the Riemann ξ function, meaning the constants being invariant, being constants conserved for all x va lues in R, for the function.

Here we elicit Pro fessor Emma Bonners Theorem, that “All cont inuous symmetries symmetries has an associated conservation conservation law”[Ref; Professor Pro fessor Kenneth W. Ford]. Hence we infer that Riemann

ξ Function is continuously symmetric giving us a clue that the Riemann ξ function funct ion must be Elliptical and the continuous symmetry must be Sine function nature of their functionality, functionality, as in Natures continuous Symmetry derived later. The continuous symmetry

of Riemann ξ function derived from Geometry and Professor

Emma Bonners theorem throws light into the similarity similar ity of it with the t he continuous continuous symmetry of  nature which is small small ellipses distributed in space space time continuum, continuum, the Granular nature of  atmosphere. This makes us elicite the Diaphentine equation y 2=x2+a which Diphentus thought had no solution but was proved by Brahma Gupta in later centuries. centuries. Brahma Gupta found that Diaphentine equation is in-fact Elliptical function and has Complex roots. We use the modern version of the so called called Elliptical Ellipt ical function of Brhama Gupta as

Differential Equation for the proof proo f of Riemann ξ function later on. The table below gives the number of Primes calculated from t he above deduction deduction of Riemann

ξ function compared to actual primes, up to 1000. One One can go beyond beyond this number. Number

Actual Primes

Calculated Calculat ed Primes from formula above

2

3

2.27

8

6

6.8

14

8

9.28

20

10

11.10

26

11

12.81

33

13

14.58

100

27

27.40

1000

170

168.75

The equation derived show that the area of o f the Riemann ξ function derived between ½ and 2 is 0, giving as an initial init ial proof for the t he trivial part of Riemann Hypothesis. Now we start the more involved proof for the same.

I start where Riemann left left his hypothesis without the proof.

n As said earlier Li(x) = 

dx/Ln(x) 2

Riemann conjured that the function ξ = ∫ 1/ln(x) has the real part of the root at ½ when s=2 We start with the time series expansion of Log function which is, ((x-1)/(x-2))+1/3((x-3)/(x-4))+1/5((x-5)/(x-6))+……..= Ln(x) Now let this be reduced reduced to the differential equation equation in the t-domain. t-domain. f ”(x)+f ’(x)+c= Ln(x)  Note(1)

Also, as a first pass check on the above equation, take, f “(x) +f’(x)+c = 1/Ln(x) Integrating we have, have, f’(x)+x+c.x= f’(x)+x+c.x=∫1/Ln(x) Considering the motion of an electron we see motion from -1 to 0 is -1, so f ‘(x)= -1 and x is motion from 0 to 1 which is 1 and substituting these numbers in the above equation we get, c=∫1/Ln(x) This also confirms that the orbits of o f an atom need be c ircular making the Riemann ξ function necessarily homogeneous in nature, where a=b in the above equation in Note(1). Prof. Albert Einstein used to repeatedly state state that t hat nature is a ‘closed form’ system and we can take it as such. The Homogeneous nature o f Riemann’s ξ function confirms it knowing the parallels of it with matter physics. Even Pauli’s exclusion principle directly shows pair existence of electrons in Molecules first orbit which has opposite ‘spins’ , the significance visible from the Riemann ξ function, which will be

more visible later on this derivations derivat ions

when we encounter encounter roots of which are conjugates. This seems to be the solution solution sought for

Poincare conjuncture. We also see later lat er that homogeneous differential equation of Riemann ξ function which is manifestation of ‘momentum’ suddenly shifts when the first t wo terms of 

the differential equation equat ion vanishes on integration to get Riemann ξ function, form one with ‘position’, an idea elusive to Quantum science even now or work is going on now which is

the significance of Paulis exclusion principle or Heisenbergs. The results of the Riemann ξ function gives the power available in a closed system when we subtract 1 from the results ie;

sum of all primes before a number which is what w hat Riemann ξ function is all about, an idea Professor Gauss might have been trying grapple with when he proposed the idea of ξ function which was again highlighted by Professor Riemann. Riemann. Also Professor Dirac’s ‘q’ numbers where transformation of Fermions to Bosons can be considered as the process taking place when the integration of 1/Ln(x) in

Riemann ξ function is done. Bosons are the carriers of o f power or transmission mechanism, so the above affirmation about power.(Ref. Smolin, Stewert) A significant grasp of Physics and and Electrical Engineering is expected expected before one can grapple

with these ideas which is my disclaimer when w hen going through the proof for Riemann ξ function.  Note(2) :

Applying Taylor series, to the time series whose 3

rd

and 2

nd

term forms of the t he equation above above

Note(1) the first two terms of the differential equation and the rest of the terms forming c, we have, 2. f “(x)/2!+ 1. f’(x)+c. f’( x)+c. Note: Taylor series is defined f(x) = Ao+A1.f’(x)/1!+A2.f”(x)/2! for a second order function and the function repeating with higher order derivatives for functions of respective orders, where A’s are the time scales, 1, 2......&C. Since, Since, higher derivatives don’t exist for lower 2

order functions. For example f(x)=x cannot be differentiated below the second differentiation. This process is initiated to elicit t he Elliptical nature of Riemann Zeta function and even if o bjection arise otherwise about the method chosen, chosen, then t hen appropriate choice of the constant term in the closed form derived nullifies such objections. Here we see a = 1 and b =1 making the log function an Elliptical function, so the roots must be Imaginary or Complex roots, which means the Integral of it’s reciprocal, also must be Elliptical and so with complex roots. Also, let

f “(x)=ω2,

f ‘ (x)=ω, where f”(x) and f’(x) are the basis of t he quadratic differential equation, which is the rate at which the axis change. change. ‘a’ , ‘b’ and ‘c’ are the t he constants which

define the function along with the basis. It is these constants that define the character of the function with basis chosen accordingly to define the dynamics. Here and later we see that the first two terms t erms of the equation vanish leaving the dynamics of the Riemann Zeta function described by the lone constant term ‘c’.

Take, ω= (-1+i.√3)/2 = 3√1, (cube root of 1) [Ref:John Derbyshire] let, a.ω2+b.ω+c=(x/ln(x))√φ, the right hand side will be revealed in a short while below. This is another form of, a.f “(x)+b.f ‘(x) + c=(x/ln(x))√φ It is known that t here are 3 cases for the above type of Quadratic differential equation where, 1)a=b 2)a>b and 3) ab and a