Riemann Hypothesis Law : Abstrtact Physics Behind Mathematical Equations

Riemann Hypothesis Commentary

13th August 2023

By Pankaj Mani

India

(manipankaj9@gmail.com)

I am not Proving Riemann Hypothesis, I am showing Riemann Hypothesis Possibly True !

There is important difference betwee the two. Former assumes Mathematics as a Theorem System, I see Mathematics as Law System !! 

Long back Sir Richard Feynman quoted that. Next great era of Human Awakening would come – Today we don’t see the content of the Equations. He was very right here in context of Riemann Hypothesis, 160 years old Mathematical puzzle, one of the most important problems in mathematics. Some of the greatest mathematical minds have unsuccessfully tried the problem. So, let’s think why is it so ? Albert Einstein said – No problem can be solved at the level that was formed. It seems to me contextually that previous attempts have been made at the same level at which Bernhard Riemann formulated the problem. There is somewhere need to look t the problem from a higher level. That’s the point in my humble view our learned Mathematician’s have not be able to do,may be due to lack of imagination. Infact it happens learned matured brained becomes too adapted often to the conventional approaches to imagine something beyond from higher level . That’s the fundamental reason, some new brainpower is required to imagine at higher level.

So, the point is – How to look at the Riemann Hypothesis from higher level perspective?

For that, we have to imagine what actually is done when a mathematician ‘s brain solves any problem. Infact what does it mean when a computer solves a problem. Infact that’s related to algorithmic approach and David Hilbert once dreamed to formalize the whole mathematics.

As Kurt Godel wonderfully demonstrated : It starts with some axioms as the base and then manipulates the axioms to prove some results subsequently by the set of sequential statements based on arithmetical operators. That’s what is called a Proof in Formal mathematics. But my humble question to the Learned Formal Mathematicians is where do those fundamental axioms come from ? Those axioms come from the day to day physical experiences of mathematicians. Those axioms are based on certain higher level physicalities.

But as David Hilbert once quoted – Advanced Mathematics is basically a Game of Symbols arbitrarily defined based on certain rules.

So, here would like to ask a very fundamental question – Is Mathematical System Self -Conscious like Human Brain which can prove themselves and those axioms upon which they are based upon ? Can a mathematical system prove those fundamental rules/axioms upon which they are built upon ? It’s like Self-referential Problem and that’s the core principle behind Gödel’s Results. Infact it’s a deeper characteristic in the Universe and Nature not just Arithmetical system as demonstrated by Gödel. Infact I tried to show that it would hold true for any mathematical system including real numbers or any system because they are not self-conscious.

So, the point is – if a problem talks about those fundamental axioms and rules, how to prove that within that mathematical system.

For example :The rules of arithmetic operations , addition, multiplication etc. have been defined based upon certain physicalities and symmetries in the Euclidean Geometrical Space .

Now If a problem comes to prove something related to those physicalities on which these Arithmetical Operations were defined, Can it be proven by using those Arithmetic Operations themselves ? That means Problem about Arithmetical Operations can’t be proven by using Arithmetical Operations only in that Mathematical system internally.

No. For that one will have to come at higher level and see the mathematical system from that perspective. That’s quite common sense.

What I mean here is that let’s say Prime Numbers are defined in our number system. But if it is asked why Prime Numbers distribution has this Physical Pattern inside their Plots. That can’t be prove by using those games of operations involving prime numbers themselves.

The reason why I have explained all these is Riemann Hypothesis somewhere is similar case. It’s related to those underlying physicalities and physical characteristics on which the rules of arithmetical operations like addition, multiple, Complex numbers etc..have been fundamentally defined. The reason in my humble view, many conventional Mathematicians are not able to solve is they are trying to solve the Self Conscious Statement of the Mathematical System by Standing within the System. As explained earlier about steps of proof, they try to do permutations and combinations of different operations internally (like playing with the piece of paper by folding it in different ways and writing on it inside) to come at the results about those external physicalities upon which those axioms about these operations were defined e.g. + multiples by + = +

(like why the Paper is a Square ?)

Like Prove that Plus multiplied by Plus = Plus using Plus & Minus themselves ? Or Prove that Circle is round ? Prove that Triangle is Triangular ? These are Self -Referential Problems.

Riemann Hypothesis is basically about looking at those Underlying Underlying Physicalities behind Mathematics itself upon which those Fundamental Symmetricity Physicalities of Arithmetical Operations, Complex Numbers, etc. were defined. If those basic Rules about Operations are changed, Riemann Hypothesis would definitely change. There is no mystery about it. The mystery is in the mind of Learned Mathematicians who are not able to imagine and look at it from higher level perspectives. First they created the system and then they are themselves finding it mysterious.

I am really worried if similar things happen during the age of AI/ML

First Mathematicians created and then they would say they find it mysterious and out of control .Infact Deep Learning etc. has become complex enough to be understood.

So, What I tried to show in my analysis to prove Riemann Hypothesis. I don’t look at the problem internally rather try to look at the Structural Symmetry & Physicalities behind the definition of those operations and variables on which Riemann Zeta Function and its Functional Equations have been created. That means Physicalities and Symmetrical Structures of Addition, Multiplication,.0, Numbers, Complex Numbers etc on which the Equations have been created !

For example, when Addition, Multiplication etc are defined on say Decimal Number System say for example

45 .5 * 20

One can see how digits are arranged at different places (1st, 10th, 100th places etc like energy orbitals where digits transition from one level (place ) to higher level(place). This structure of Number systems and the Operations itself is borrowed from Quantum Energy Orbitals like for Electrons at different energy levels.

My point is that behind all these defined algebraic and arithmetic structures in Mathematics exist the Physicalities based on certain Symmetries.

What does “0” represent ? It lies on the midpoint line of symmetry on the Number line.

The Point I’ve been trying to convey that certain physicalities of symmetry lies behind the scene of these mathematical operations and numbers and functions.

So, I tried to look at the Symmetrical Physical Structure of the Equations and Relate Correspondingly to its Physical Graph and how in this Game of Symbols, such foundational underlying Physicalities and Symmetricity will have to remain Conserved.

Like if the Algebraic Equation of a Circle is Symmetric and Homogeneous, the Physicality behind its graph will also remain Symmetric and Homogeneous. If we tilt the Equation of Circle to form the Equation of Ellipse or something else, its physicalities would also change correspondingly.

On that basis as David Hilbert quoted mathematics being the game of symbols. Intrepid to play the game while conserving those symmetric physicalities. That’s sufficient to prove Riemann Hypothesis for Riemann Zeta function.

I’ll explain the proof here.

This is Riemann Zeta Function for Re(s) >1

And its analytic continuation elsewhere.

The Functional Equation Satisfied is

We just look at the functional equation when the LHS term can be 0. Also one knows that many other power terms and Gamma function never attains the 0 value, so eventually it fturns out to be a simple functional equation of the form

f(s)=sin(π*s\/2)*f(1-s)f(1/2-s

With some transformation s replaced by ½ - s ,

It becomes

f(1\/2 - s) = sin[(π*1\/2*(1\/2-s)] *f(1\/2+s)

So, It’s the game of three terms

 f( 1\/2-s) on LHS & sin( ) & f(1\/2 +s) on RHS

Now I apply the Rules of Multiplication of 0 to find out when f(┤) can be 0. The Trajectory of Trivial 0s already come from the same equation. The Trajectory of Non-Trivial 0s would also come from the same equation.

So, what I did, I visualized this as the game of these three terms and showed that only way to arrange the Non-Trivial 0s would be when they lie on the critical line or else the entire function would be 0. But this symmetry in the graph would be true for Riemann Zeta function only because of the symmetry in the structure of equations of RZ function. All other Counter examples like Finite Sum of Dirichlet L functions or many others will not be as symmetric and homogenous as the Riemann Zeta function in terms of the structure and arrangement of mathematical symbols. This is where one can imagine what I told that Equation of Circle being Symmetric and Homogenous is the reason why the Graph of Circle also has Symmetry and Homogeneity. If there is introduced some asymmetry and heterogeneity in the equation of circle, say like ellipse or something else, the graphical representation also gets similar asymmetry and heterogeneity.

One needs to look at the similar symmetry in the Structural Equation of RZ function as well and that’s the reason why all the values of s when sin () =0 and when sin () is not ), they behave homogeneously and symmetrically for RZ function in the arrangement of 0s game among those three functions. For other Counter examples, one can find out how different symbols lack the distortion in the symmetry and homogeneity leading to distortion in the graphs and hence possible violation of the Non-trivial Zeros being on the Line of Symmetry. Critical Line is basically the Line of Symmetry just like 0 lies on the line of symmetry of the Number line.

If someone says that Non trivial 0s are not on critical line say they are on Re(s) = ½-s and ½ +s for some specific value of s. then my question on the basis of symmetry and homogeneity would be when there was no asymmetricity introduced in the game while defining them why it will be asymmetric for some specific value of s and not others ? Why this asymmetricity would occur if the RZ function is symmetric and homogeneous in the structure of its equation? This is where the imagination is required to be able to look inside the structure of the equation and corresponding structure in the graph.

So, my So Called Law(in place of Theorem is) : Physicalities of Symmetry and Asymmetry assumed at the foundation of axioms and rules behind the definition of mathematical/ arithmetical/algebraic system/game definitions remains conserved in the graphical form as well !

Like Emmy Noether ‘s Theorems based on Symmetry and Conservation Laws in Physical System, My point is the Conservation of Physicalities behind Mathematics.

Even Mathematics has its own abstract Physics like the Physics of Bodies in Real World. So, new branch of Mathematics like Arithmetic Physics or similar should study this abstract Physics of Mathematics itself !

Hence rather than making mathematics as a mechanical system of theorems and axioms, we should further study it as a discipline like Laws of Mathematics where the Underlying abstract Physics of Structure of Equations, Graphs etc are deeply studied !!

Infact this will lead to new branch of mathematics at the boundary of mathematics and physics where symmetrical rules behind the mathematical system’s definition needs to be studied in detail for further advancement !

It also paves the way forward to broaden the new branch of mathematics called Arithmetical Physics or Some other Physics where those Physicalities behind the Basic Rules of Mathematical Systems are studied deeply. I had talked about these things and hidden concepts in my paper around 2011(publicly available ) and then subsequently as well as an amateur Number Theorist . Sir Michael Atiyah approach to the Problem (2018) talks about similar Physicalities( Arithmetic Physics) to some extent .

What is the need of the hour that our learned Contemporary Mathematicians need to broaden their views of Mathematics rather than just paying Permutations and Combinations of those game rules like a machine which even Computer can play to some extent. But Mathematics is beyond that.. It’s not Self-conscious to prove the results about those Physicalities behind the Rules/Axioms upon which they are formulated. Here is the Role of Human Conscious Understanding of the Mathematicians’ brains. Simply by making a mathematical tough and tough by maintaining the inertia that they don’t have to solve every problem by residing at the same level on which they are formed as Albert Einstein wonderfully quoted , would let them go nowhere except maintaining and satisfying their ego for centuries at the cost of future development of true beauty of mathematics as the creative subject rather a mechanical subject !!

 Mathematicians make it tough by not going at higher level of imagination rather solving at lower level

 Physicalities of Symmetry remains conserved if the algebraic equation has symemtry.

Riemann Hypothesis Analysis :Mystery Hidden in the Structure of Equations itself !! Physics behind Mathematical Arithmetical Operations.

Riemann Hypothesis Resolution using Game theory

Note : This paper is based on my discussion with Prof .Ken Ono, Celebrated Number Theorist Professor University of Virginia)(Former VP at American Mathematical Society) way back in 2019 .This paper was written in 2019 and before when I saw it few days ago and thought to publish it publicly.

Abstract :

Riemann Hypothesis is TRUE if we look at the Functional Equation satisfied by the Riemann Zeta function upon analytical continuation in Game Perspective way as visualized by David Hilbert. The functional equation already shows the existence of trivial zeros . Here, in this paper I try to use the same functional equation to find out the location of non-trivial zeros and hence show that Riemann hypothesis is true for Riemann Zeta function. It uses technical game theoretical concept of Nash Equilibrium. There is need to imagine the Foundational Principles underlying Mathematics . In other words, it’s the game of arranging Zeros on the complex plane using the functional equation.

The basic idea is that Richard Feynman once told : We don’t see the content of Equations. I believe naturally that Symmetry, asymmetry in the structure of algebraic equations lead to similar physical characteristics in their graphs as well and vice versa. Mathematical systems like a game as demonstrated by David Hilbert based on certain axioms derived from the Mathematician’s experiences of physical world can’t possibly prove those physicalities of the axioms itself by playing with the symbols. For example : One can’t possibly prove the fundamental physicalities of the arithmetic Operators Addition, Multiplication etc using these Operators themselves. The Physicalities remain conserved while doing these Artihmetics ! Hence to show them one has to look at the mathematical system being an external observer rather trying to do by being internal to the system . It’s like trying to lift a bucket up by standing inside it !!

Riemann Hypothesis is about looking at the mathematical equations as an external observer to the system, explore its symmetry and asymmetry and link with the symmetry and asymmetry in physicalities. Like for example a circle having symmetric homogeneous algebraic equations have similar characteristics in its graphical pictures as well. There can’t be mismatch the way the fundamental rules of the mathematical systems here are defined !! RH is about the whole Mathematics where one has to reimagine the mathematics as an external observer in my humble view.

Physicalities Conservarion behind the Arithmetical Operations.. Physics behind Arithmetical Operations . That's also the Physics behind Godel's undecidable results 

INTRODUCTION:

In this paper, I will be looking functional equation satisfied by Riemann zeta function actually a non-cooperative game between its constituent terms(here different mathematical functional symbols) in which the best strategy adopted by each player to locate zeros on mathematical field leads to discovering the most stable arrangement of physical location non-trivial zeros of Riemann zeta function, which in turn leads to TRUTHFULNESS OF RIEMANN HYPOTHESIS..

As visualized by David Hilbert- Mathematics is actually a game between different mathematical symbols, where different symbols follow certain defined rules.

The mathematical theory of games was invented by John von Neumann and Oskar Morgenstern (1944).Game theory is the study of the ways in which strategic interactions among agents produce outcomes with respect to the preferences (or utilities) of those agents, where the outcomes in question might have been intended by none of the agents.. All situations in which at least one agent can only act to maximize his utility through anticipating (either consciously, or just implicitly in his behavior) the responses to his actions by one or more other agents is called a game. Agents involved in games are referred to as players. If all agents have optimal actions regardless of what the others do, as in purely parametric situations or conditions of monopoly or perfect competition we can model this without appeal to game theory; otherwise, we need it.

Each player in a game faces a choice among two or more possible strategies. A strategy is a predetermined ‘programme of play’ that tells her what actions to take in response to every possible strategy other players might use. I will prominently use the tools of game theory to find out different Nash equilibrium stage in this functional game played between mathematical symbols.

Here, in particular, I visualize the functional equation satisfied by Riemann zeta function as game between different constituent terms which are connected through multiplication sign on both side of equality sign.. I would be finding the Nash Equilibrium which will be the solution and prove the Riemann Hypothesis to be True.

. As this has exactly 1 NE stage corresponding to the location of non-trivial zeros on the critical line in 0<R(s) <1.

So, what I would be doing is- finding the locations of trivial & non-trivial zeros by looking the arithmetic structure of Riemann zeta function and by applying the two basic arithmetic of numeric ‘0’ to find out different set of possibilities of taking zero value by different constituent terms.

In a nutshell, I will NOT go into finding the zeros of this functiuon, rather I will be visualizing the arithmetic structure of FUNCTIONAL EQUATION ,in which different constituent terms are connected through multiplicative sign and using game theory find the NE stage to locate zeros. So, it has hardly anything to do with anything else than game theory and arithmetic of numeric 0.

The Riemann zeta function ((s) is a function of a complex variable s = o + it (here, s, o and t are traditional notations associated to the study of the Ç-function). The following infinite series converges for all complex numbers s with real part greater than 1, and defines (s) in this case:

The Riemann zeta function is defined as the analytic continuation of the function defined for o > 1 by the sum of the preceding series.

The Riemann zeta function satisfies the functional equation

where T(s) is the gamma function which is an equality of meromorphic functions valid on the whole complex plane. This equation relates values of the Riemann zeta function at the points s and 1 – s. The gamma function has a simple pole at every non-positive integer, therefore, the functional equation implies that ((s) has a simple zero at each even negative integer s = – 2n Pi these are the trivial zeros of ((s).

Incidentally, this relation is interesting also because it actually exhibits ((s) as a Dirichlet series (of the y-function) which is convergent (albeit non-absolutely) in the larger half- plane o > 0 (not just o > 1), up to an elementary factor.

Statement of Riemann Hypothesis:

All non-trivial zeros of Riemann zeta function in the critical space 0<R(s)<1 lies on R(s)=1/2 .

Here we look at Game theoretic aspects of how to arrange the Zeros on this plane.

, I visualize numbers and their mathematical functions playing the game of symbols .

In context of functional equation game played by Riemann zeta functions in the game there are two players A & B where A corresponds to sin() NOT =0 and B corresponds to sin() =0.

A solution concept in game theory :

Nash Equilibrium which corresponds to the solution,here the physical location of non-trivial zeros of Riemann zeta function.

PROOF:

Functional equation satisfied by Players Ç (s) & Ç(1-s) in the entire complex domain ‘C’ is

As one and only one term on each side of “=” sign can and must be zero as 0*0 = 0 &

0 *non-zero number= 0

2^s(Pi)^(s-1) and Gamma function terms can never be equal to 0 ,so we can skip that here as they will not contribute to becoming 0 using the functional equation.

And by coordinate transformation, s & 1-s can be transformed to ½-s and ½+s

Note: I have used “f(s)” in place of the function in the Riemann Zeta functional equation further for simplicity.

Notations

• A = Function f(s) for { C: s :: sin(Pi*s/2 )=0,s‡0} as s=0 is the location for pole

i.e. those values of s for which Sin(Pi*s/2) is not equal to 0.

• B = Function f(s) for { C-A,s‡0} i.e. those values of s for which Sin(Pi*S/2) =0

Player A (for which Sin () term is not 0) has also two options .It can also exercise one of the two.

1. Ç(s)=0 for R(s)> 1/2 and simultaneously for R(s)<1/2 (Both sides 0 simultaneously)

2. Ç(s)=0 for s= ½+it But, Ç(s)‡ 0 for R(s)>1 ( Or none of the sides will be 0)

i.e. C(s) ‡0 for R(s)<1/2 and C(s) ‡0 for R(s) >1/2

Similarly,

Player B (for which Sin() term =0) has two options to exercise in the game .It can exercise only one of the two.

1. Ç(s) =0 for R(s)>1/2, Ç(s)= 0 for R(s)<1/2 i.e.(Both sides will be 0)

2. Ç(s)=0 for R(s)<1/2, Ç(s)‡0 for R(s)>1/2 (Left side of R(s)=1/2 will be Zero ,Right side will not be zero)

Now, we look at the different permutations of strategies adopted in this game and find their payoff matrix.

 Payoff matrix of this game for the Riemann Zeta function

 Player A exercises

1st option Player A exercises

2nd option

Player B exercises

1st option 0,0(All the points 0)Impossible as it means f(s)=0 for all s 0,0 (0 on both sides for Sin()=0 Impossible because it is already proven that there are no zeros for R(s)>1 for Riemann Zeta function.

Player B exercises

2nd option 0,0(All one side points = 0) Impossible as only trivial zeros already known. Impossible 1,1 (Possible location for 0) The only possible way to gain the stability and maximizes the payoff. Equilibrium Stage for Riemann Zeta function.

By looking at the table Payoff is maximum i.e.(1,1) when A exercises 2nd and B also exercises 2nd option to locate Zeros.

The players A & B (the sub players derived from Sine function) both similarly exercise their respective options uniformly.

That’s the Nash equilibrium state by looking when both the players exercise the 2nd options.

Which means that f(s) =0 in the critical strip 0< R(s)<1/2 will not exist either on the left side of R(s)=1/2 nor right side. So, the only possible location for the Non-Trivial Zeros would be R(s)=1/2 for Riemann Zeta function.

.

This asserts the truthfulness of the Riemann hypothesis for Riemann Zeta function that trivial zeros lie on the points s=2k,k<0 and non-trivial zeros will lie on the R(s)=1/2 .Thus,

It implies that

Ç(s) =0 for R(s)=1l2+it for 0<R(s)<1 and also Ç(s) ‡ 0 for R(s)>1/2

QED

References:

1. GAME THEORY http://plato.stanford.edu/entries/game-theory/#Games

2. Mathematics as Game https://www.marxists.org/reference/subject/philosophy/works/ge/hilbert.htm

3. Riemann hypothesis by Enrico Bombieri

 https://www.issnaf.org/inside-issnaf/enrico-bombieris-lectures-on-the-riemann-zeta-functions.html

Explanations to the possibilities of Counterexamples as demonstrated (Those functions satisfying the same one variable Riemann zeta functional equation but Riemann Hypo. NOT TRUE for them) in terms of the strategies in the game.

Here I am showing that there may exist possibilities for the counter examples in this above game based under constraints i.e. cooperative/coalition behavior of the variable within the game. In such cases, the strategy will be different in the same game.

The one variable generic functional equation found for Riemann Zeta function implies to the fact Riemann Hypothesis is TRUE. But there may exist some counter examples to this as pointed out by Prof.Ken Ono.

In any case of counter examples, I have shown below the possible type of strategy in the same game that will be followed by those specific functions whether L-functions or any other type of functions in general. There may be many such functions as counterexamples. One has to study those functions deeply separately to find out the sub players classification details in that strategy for the counter examples.

If certain functions e.g L-functions (two-variables) or other functions are forcibly made to satisfy the same one variable functional equation, this leads to the external constraints in the functional equation game and changes the fundamental aspect of the game from non-cooperative to cooperative under external constraints.. As a result of this, the strategies in the same game will differ from the original basic one derived for the (Riemann Zeta) RZ function. This enforces constraints in terms of other variables/parameters/other aspects on the behavior of the function for the variable.

[Kindly note that the two variables generic functional equations involving S and Dirichlet parameters X(n) for L-functions is different without any constraints.]

Coming to the discussion of the strategies for the counterexamples mentioned above. There could be possible many counter examples when there exist some external constraints in terms of variable, parameters or regrouping of some parts etc. . In that case there will be added sub players in the game for the variable s depending upon the external constraint/variable say of L-function or any other counter example functions.

Let me explain the possible case just in context of the counterexample

Player A : When sin() not equals 0, there are two options originally if f(1/2-s) =0, then f(1/2+s) also equals 0

Or if f(1/2-s) Not equals 0, then f(1/2+s) also Not equals 0.

But the external constraints leads to subplayers for the player A & B for variable S namely say A(a),A(b) & B(a),B(b).

When sin() not equals 0,

 For the subplayer A(a),

  F(1/2-s) =0 and f(1/2+s)=0 and

  For the sub player A(b)

 (1/2-s) not equals 0 and f(1/2+s) not equals 0.

In this case Riemann Hypothesis may NOT necessarily be TRUE!

Now the sub players A(a),A(b),B(a)& B(b) for the variable S will depend upon the various counter examples-. This needs to be discovered for each counterexample case separately. Hence the different sub players (i.e. different group of values of s ) will follow the strategies.

Euler product form based counter examples : Everything comes into the functional equation. I am completely looking at the one variable generic functional equation derived and satisfied for Riemann Zeta function. Euler product contains another function called multiplicative functions, which could lead to a constraint . If some of the terms of the Euler product form are modified externally to satisfy the one variable RZ functional equation to produce some counter examples , it will create new sub players for the player variable S depending upon the coalition characteristics and behavior of the modified terms and hence will enforce external constraint. So again different strategies will arise for different subplayers in the game. The generic functional equation satisfying those modified functions without any constraint would be different. The entire set of sub players and the generic functional equations satisfied requires to be found on case to case basis for various counterexamples.

 So, then the solution i.e. the equilibrium point will be decided upon considering the strategies of sub players. This is infact technically cooperative game where due to external constraints, the formation of coalition for sub players is formed and in that case the equilibrium and solution is calculated by taking the various combinations of coalition of sub players. But in that case as the strategies will be different for sub players , it can lead to the violation of Riemann Hypothesis Truthfulness as all the non-trivial 0s will not lie on the critical line R(s)=1/2 because of possibility of one more sub option where both the f(1/2-s) and f(1/2+s) becomes 0 when sin() not equals 0.

To summarize , under external constraints and cooperative behavior of sub-players of the variable due to modifications or whatsoever, the case of counter examples becomes a case of further cooperative subgames and will be dealt accordingly separately. But in that case RH needs not be TRUE.

In the list of strategies mentioned below, I am showing the strategy within the game for all the counter examples in general .

Notations for the Sub players.

Player A will now have two sub players A(a) & A(b) based on the characteristic of their behavior in variable S with each having two options.

Similarly, Player B will now have two sub players B(a) & B(b) with each having two options

Now various combinations of Payoff Matrix of the Game under Constraints would be like this as follows: I am showing the possible type of strategy against Riemann hypothesis.(i.e. It may NOT be True).

##The strategies of the Game and their Payoffs as follows:

1)A (a) 1st option + A(b) 1st option + B(a) 1st option +B(b) 1st option

( In this case f(s)=0 ,impossible)

2) A(a) 1st option +A(b)1st option + B(a)2nd option +B(b)2nd option

In this case all one side to the left f(s)=0,impossible.)

3)A(a) 2nd option + A(b)2nd option +B(a) 1st option +B(b)1st option

(In this case Both sides 0, RH may be True for some other functions apart from Riemann zeta function also but impossible for Riemann Zeta function as shown in the matrix payoff earlier as it has no trivial 0s for R(s)>1).

4)A(a)2nd option+ A(b) 2nd option + B(a)2nd option+ B(b)2nd option

 (In this case RH True for Riemann Zeta function as shown earlier in the matrix payoff in the original paper)

5)A(a)1 option +A(b) 2nd option + B(a) 1st option +B(b) 2nd option.

 (In this case RH may not be true as for various counter examples)

Hence, the last one is the possible strategy corresponding to the various counter examples satisfying the one variable Riemann Zeta functional Equation but violating Riemann hypothesis.

So, the functional equation game shows that Riemann hypothesis will be True for Riemann Zeta function and some other functions but may NOT be TRUE for various counterexamples functions.

Assumptions & Approximation

 Assumptions and Approximations could be more dangerous than Nuclear Bombs !!

Causal Based Learning Algorithm in AI/ML for Finance

My New type of Learning Algorithm in AI/ML in Finance : Causal Based Learning which would learn based on Causality Principle combined with Randomized approaches like Reinforcement Learning 

Symmetricity : Riemann Hypothesis

The problem at the heart of mathematics and physics. 

So some brief background what I was taking to you last time. I try to look at the mathematical system here as a game where one has defined the rules for operators like addition, multiplication, subtraction, division etc. There can be made different games by tweaking the rules of algebra and numbers. Probably that’s the reason why David Hilbert looked at advanced maths as the game of symbols. 

Since, it’s the game I tried to look at the physical structure behind the RZ function on he basic of symmetry aspects in the game.

Last time I showed that Riemann Zeta function Satisfies the Functional Equations and by using Functional Equation I tried to show why RH is true . But then you came back with wonderful counter examples to that using Dirichlet L functions , Polya etc. 

Then I also reverted after my thoughts that somewhere I am trying to link Symmetricity in the structure of Equations and the Graphical Plots. By Structural Symmetricity in the Equation of RZ, I tried to relate that to the game rules and also conservation of physicalities in the graph where 0 represents the singularity and physicalities is collinearity of non trivial zeros too. 

I then found that these counter examples have asymmetric aspects unlike Riemann zeta function in their structures and hence their game theory strategies would get distorted and hence also the graphical characteristics would likely be asymmetric and that’s why these counter examples might violate RH due to asymmetry and heterogenous in their structure of equations. 

On that basis, as per the Game theory rule or physical characteristics conservation, I tried to show that RH would be true for RZ function but may not be true for other Counter examples. There can be created many Counter examples but not as symmetric as RZ.

On that intuition and principle I tried to show RH is true for RZ ! I tried to conserve the symemtricity that were assumed while defining the arithmetic and algebraic rules of addition, multiplication, subtraction etc.. and establish the link between the symmetricity,homogeneity in algebraic equations and the graphical representation like for example the equation of circle is symmetric and homogeneous in algebraic structure and hence also symmetric and homogeneous in graphical representation.similarly for RZ function and Other Counter Examples satisfying the RZ functional Equations. I don’t see that an asymmetric equations will have graphical symmetry and symmetric equations curve will have asymmetric graphical structure !! That’s my main point.

So, I would kindly request to have some imaginative thoughts on my humble idea and reimagine Mathematics from a different angle. 

(I am trying to look from Natural Point of view not from Conventional approaches employed by expert number theorist. Because I see some possible fundamental limitations there . It’s like trying to lift the bucket by standing inside the bucket which won’t be possible !! 

This is because I believe that in mathematics one can prove many things but may not prove the axioms itself on which they have been defined. For that one has to possibly imagine the mathematical problem by treating it like a physical system externally as an observer not internally using arranging equations. 

Let's discuss mathematics unconventionally..

 

Riemann Hypothesis: Game Theoretic Proof : Reimagining Mathematics

Riemann Hypothesis Resolution using Game theory : Reimagination of Mathematics 

Note : This paper is based on my discussion with Prof .Ken Ono, Celebrated Number Theorist Professor University of Virginia)(Former VP at American Mathematical Society) way back in 2019 .This paper was written in 2019 and before when I saw it few days ago and thought to publish it publicly.

Abstract :

Riemann Hypothesis is TRUE if we look at the Functional Equation satisfied by the Riemann Zeta function upon analytical continuation in Game Perspective way as visualized by David Hilbert. The functional equation already shows the existence of trivial zeros . Here, in this paper I try to use the same functional equation to find out the location of non-trivial zeros and hence show that Riemann hypothesis is true for Riemann Zeta function. It uses technical game theoretical concept of Nash Equilibrium. There is need to imagine the Foundational Principles underlying Mathematics . In other words, it’s the game of arranging Zeros on the complex plane using the functional equation.

INTRODUCTION:

In this paper, I will be looking functional equation satisfied by Riemann zeta function actually a non-cooperative game between its constituent terms(here different mathematical functional symbols) in which the best strategy adopted by each player to locate zeros on mathematical field leads to discovering the most stable arrangement of physical location non-trivial zeros of Riemann zeta function, which in turn leads to TRUTHFULNESS OF RIEMANN HYPOTHESIS..

As visualized by David Hilbert- Mathematics is actually a game between different mathematical symbols, where different symbols follow certain defined rules.

The mathematical theory of games was invented by John von Neumann and Oskar Morgenstern (1944).Game theory is the study of the ways in which strategic interactions among agents produce outcomes with respect to the preferences (or utilities) of those agents, where the outcomes in question might have been intended by none of the agents.. All situations in which at least one agent can only act to maximize his utility through anticipating (either consciously, or just implicitly in his behavior) the responses to his actions by one or more other agents is called a game. Agents involved in games are referred to as players. If all agents have optimal actions regardless of what the others do, as in purely parametric situations or conditions of monopoly or perfect competition we can model this without appeal to game theory; otherwise, we need it.

Each player in a game faces a choice among two or more possible strategies. A strategy is a predetermined ‘programme of play’ that tells her what actions to take in response to every possible strategy other players might use. I will prominently use the tools of game theory to find out different Nash equilibrium stage in this functional game played between mathematical symbols.

Here, in particular, I visualize the functional equation satisfied by Riemann zeta function as game between different constituent terms which are connected through multiplication sign on both side of equality sign.. I would be finding the Nash Equilibrium which will be the solution and prove the Riemann Hypothesis to be True.

. As this has exactly 1 NE stage corresponding to the location of non-trivial zeros on the critical line in 0<R(s) <1.

So, what I would be doing is- finding the locations of trivial & non-trivial zeros by looking the arithmetic structure of Riemann zeta function and by applying the two basic arithmetic of numeric ‘0’ to find out different set of possibilities of taking zero value by different constituent terms.

In a nutshell, I will NOT go into finding the zeros of this functiuon, rather I will be visualizing the arithmetic structure of FUNCTIONAL EQUATION ,in which different constituent terms are connected through multiplicative sign and using game theory find the NE stage to locate zeros. So, it has hardly anything to do with anything else than game theory and arithmetic of numeric 0.

The Riemann zeta function ((s) is a function of a complex variable s = o + it (here, s, o and t are traditional notations associated to the study of the Ç-function). The following infinite series converges for all complex numbers s with real part greater than 1, and defines (s) in this case:

The Riemann zeta function is defined as the analytic continuation of the function defined for o > 1 by the sum of the preceding series.

The Riemann zeta function satisfies the functional equation

where T(s) is the gamma function which is an equality of meromorphic functions valid on the whole complex plane. This equation relates values of the Riemann zeta function at the points s and 1 – s. The gamma function has a simple pole at every non-positive integer, therefore, the functional equation implies that ((s) has a simple zero at each even negative integer s = – 2n Pi these are the trivial zeros of ((s).

Incidentally, this relation is interesting also because it actually exhibits ((s) as a Dirichlet series (of the y-function) which is convergent (albeit non-absolutely) in the larger half- plane o > 0 (not just o > 1), up to an elementary factor.

Statement of Riemann Hypothesis:

All non-trivial zeros of Riemann zeta function in the critical space 0<R(s)<1 lies on R(s)=1/2 .

Here we look at Game theoretic aspects of how to arrange the Zeros on this plane.

, I visualize numbers and their mathematical functions playing the game of symbols .

In context of functional equation game played by Riemann zeta functions in the game there are two players A & B where A corresponds to sin() NOT =0 and B corresponds to sin() =0.

A solution concept in game theory :

Nash Equilibrium which corresponds to the solution,here the physical location of non-trivial zeros of Riemann zeta function.

PROOF:

Functional equation satisfied by Players Ç (s) & Ç(1-s) in the entire complex domain ‘C’ is

As one and only one term on each side of “=” sign can and must be zero as 0*0 = 0 &

0 *non-zero number= 0

2^s(Pi)^(s-1) and Gamma function terms can never be equal to 0 ,so we can skip that here as they will not contribute to becoming 0 using the functional equation.

And by coordinate transformation, s & 1-s can be transformed to ½-s and ½+s

Note: I have used “f(s)” in place of the function in the Riemann Zeta functional equation further for simplicity.

Notations

• A = Function f(s) for { C: s :: sin(Pi*s/2 )=0,s‡0} as s=0 is the location for pole

i.e. those values of s for which Sin(Pi*s/2) is not equal to 0.

• B = Function f(s) for { C-A,s‡0} i.e. those values of s for which Sin(Pi*S/2) =0

Player A (for which Sin () term is not 0) has also two options .It can also exercise one of the two.

1. Ç(s)=0 for R(s)> 1/2 and simultaneously for R(s)<1/2 (Both sides 0 simultaneously)

2. Ç(s)=0 for s= ½+it But, Ç(s)‡ 0 for R(s)>1 ( Or none of the sides will be 0)

i.e. C(s) ‡0 for R(s)<1/2 and C(s) ‡0 for R(s) >1/2

Similarly,

Player B (for which Sin() term =0) has two options to exercise in the game .It can exercise only one of the two.

1. Ç(s) =0 for R(s)>1/2, Ç(s)= 0 for R(s)<1/2 i.e.(Both sides will be 0)

2. Ç(s)=0 for R(s)<1/2, Ç(s)‡0 for R(s)>1/2 (Left side of R(s)=1/2 will be Zero ,Right side will not be zero)

Now, we look at the different permutations of strategies adopted in this game and find their payoff matrix.

 Payoff matrix of this game for the Riemann Zeta function

 Player A exercises

1st option Player A exercises

2nd option

Player B exercises

1st option 0,0(All the points 0)Impossible as it means f(s)=0 for all s 0,0 (0 on both sides for Sin()=0 Impossible because it is already proven that there are no zeros for R(s)>1 for Riemann Zeta function.

Player B exercises

2nd option 0,0(All one side points = 0) Impossible as only trivial zeros already known. Impossible 1,1 (Possible location for 0) The only possible way to gain the stability and maximizes the payoff. Equilibrium Stage for Riemann Zeta function.

By looking at the table Payoff is maximum i.e.(1,1) when A exercises 2nd and B also exercises 2nd option to locate Zeros.

The players A & B (the sub players derived from Sine function) both similarly exercise their respective options uniformly.

That’s the Nash equilibrium state by looking when both the players exercise the 2nd options.

Which means that f(s) =0 in the critical strip 0< R(s)<1/2 will not exist either on the left side of R(s)=1/2 nor right side. So, the only possible location for the Non-Trivial Zeros would be R(s)=1/2 for Riemann Zeta function.

.

This asserts the truthfulness of the Riemann hypothesis for Riemann Zeta function that trivial zeros lie on the points s=2k,k<0 and non-trivial zeros will lie on the R(s)=1/2 .Thus,

It implies that

Ç(s) =0 for R(s)=1l2+it for 0<R(s)<1 and also Ç(s) ‡ 0 for R(s)>1/2

QED

References:

1. GAME THEORY http://plato.stanford.edu/entries/game-theory/#Games

2. Mathematics as Game https://www.marxists.org/reference/subject/philosophy/works/ge/hilbert.htm

3. Riemann hypothesis by Enrico Bombieri

 https://www.issnaf.org/inside-issnaf/enrico-bombieris-lectures-on-the-riemann-zeta-functions.html

Explanations to the possibilities of Counterexamples as demonstrated (Those functions satisfying the same one variable Riemann zeta functional equation but Riemann Hypo. NOT TRUE for them) in terms of the strategies in the game.

Here I am showing that there may exist possibilities for the counter examples in this above game based under constraints i.e. cooperative/coalition behavior of the variable within the game. In such cases, the strategy will be different in the same game.

The one variable generic functional equation found for Riemann Zeta function implies to the fact Riemann Hypothesis is TRUE. But there may exist some counter examples to this as pointed out by Prof.Ken Ono.

In any case of counter examples, I have shown below the possible type of strategy in the same game that will be followed by those specific functions whether L-functions or any other type of functions in general. There may be many such functions as counterexamples. One has to study those functions deeply separately to find out the sub players classification details in that strategy for the counter examples.

If certain functions e.g L-functions (two-variables) or other functions are forcibly made to satisfy the same one variable functional equation, this leads to the external constraints in the functional equation game and changes the fundamental aspect of the game from non-cooperative to cooperative under external constraints.. As a result of this, the strategies in the same game will differ from the original basic one derived for the (Riemann Zeta) RZ function. This enforces constraints in terms of other variables/parameters/other aspects on the behavior of the function for the variable.

[Kindly note that the two variables generic functional equations involving S and Dirichlet parameters X(n) for L-functions is different without any constraints.]

Coming to the discussion of the strategies for the counterexamples mentioned above. There could be possible many counter examples when there exist some external constraints in terms of variable, parameters or regrouping of some parts etc. . In that case there will be added sub players in the game for the variable s depending upon the external constraint/variable say of L-function or any other counter example functions.

Let me explain the possible case just in context of the counterexample

Player A : When sin() not equals 0, there are two options originally if f(1/2-s) =0, then f(1/2+s) also equals 0

Or if f(1/2-s) Not equals 0, then f(1/2+s) also Not equals 0.

But the external constraints leads to subplayers for the player A & B for variable S namely say A(a),A(b) & B(a),B(b).

When sin() not equals 0,

 For the subplayer A(a),

  F(1/2-s) =0 and f(1/2+s)=0 and

  For the sub player A(b)

 (1/2-s) not equals 0 and f(1/2+s) not equals 0.

In this case Riemann Hypothesis may NOT necessarily be TRUE!

Now the sub players A(a),A(b),B(a)& B(b) for the variable S will depend upon the various counter examples-. This needs to be discovered for each counterexample case separately. Hence the different sub players (i.e. different group of values of s ) will follow the strategies.

Euler product form based counter examples : Everything comes into the functional equation. I am completely looking at the one variable generic functional equation derived and satisfied for Riemann Zeta function. Euler product contains another function called multiplicative functions, which could lead to a constraint . If some of the terms of the Euler product form are modified externally to satisfy the one variable RZ functional equation to produce some counter examples , it will create new sub players for the player variable S depending upon the coalition characteristics and behavior of the modified terms and hence will enforce external constraint. So again different strategies will arise for different subplayers in the game. The generic functional equation satisfying those modified functions without any constraint would be different. The entire set of sub players and the generic functional equations satisfied requires to be found on case to case basis for various counterexamples.

 So, then the solution i.e. the equilibrium point will be decided upon considering the strategies of sub players. This is infact technically cooperative game where due to external constraints, the formation of coalition for sub players is formed and in that case the equilibrium and solution is calculated by taking the various combinations of coalition of sub players. But in that case as the strategies will be different for sub players , it can lead to the violation of Riemann Hypothesis Truthfulness as all the non-trivial 0s will not lie on the critical line R(s)=1/2 because of possibility of one more sub option where both the f(1/2-s) and f(1/2+s) becomes 0 when sin() not equals 0.

To summarize , under external constraints and cooperative behavior of sub-players of the variable due to modifications or whatsoever, the case of counter examples becomes a case of further cooperative subgames and will be dealt accordingly separately. But in that case RH needs not be TRUE.

In the list of strategies mentioned below, I am showing the strategy within the game for all the counter examples in general .

Notations for the Sub players.

Player A will now have two sub players A(a) & A(b) based on the characteristic of their behavior in variable S with each having two options.

Similarly, Player B will now have two sub players B(a) & B(b) with each having two options

Now various combinations of Payoff Matrix of the Game under Constraints would be like this as follows: I am showing the possible type of strategy against Riemann hypothesis.(i.e. It may NOT be True).

##The strategies of the Game and their Payoffs as follows:

1)A (a) 1st option + A(b) 1st option + B(a) 1st option +B(b) 1st option

( In this case f(s)=0 ,impossible)

2) A(a) 1st option +A(b)1st option + B(a)2nd option +B(b)2nd option

In this case all one side to the left f(s)=0,impossible.)

3)A(a) 2nd option + A(b)2nd option +B(a) 1st option +B(b)1st option

(In this case Both sides 0, RH may be True for some other functions apart from Riemann zeta function also but impossible for Riemann Zeta function as shown in the matrix payoff earlier as it has no trivial 0s for R(s)>1).

4)A(a)2nd option+ A(b) 2nd option + B(a)2nd option+ B(b)2nd option

 (In this case RH True for Riemann Zeta function as shown earlier in the matrix payoff in the original paper)

5)A(a)1 option +A(b) 2nd option + B(a) 1st option +B(b) 2nd option.

 (In this case RH may not be true as for various counter examples)

Hence, the last one is the possible strategy corresponding to the various counter examples satisfying the one variable Riemann Zeta functional Equation but violating Riemann hypothesis.

So, the functional equation game shows that Riemann hypothesis will be True for Riemann Zeta function and some other functions but may NOT be TRUE for various counterexamples functions.

The problem at the heart of mathematics and physics. 

I try to look at the mathematical system here as a game where one has defined the rules for operators like addition, multiplication, subtraction, division etc. There can be made different games by tweaking the rules of algebra and numbers. Probably that’s the reason why David Hilbert looked at advanced maths as the game of symbols. 

Since, it’s the game I tried to look at the physical structure behind the RZ function on he basic of symmetry aspects in the game.

I showed that Riemann Zeta function Satisfies the Functional Equations and by using Functional Equation I tried to show why RH is true . But then you came back with wonderful counter examples to that using Dirichlet L functions , Polya etc. 

Then I also reverted after my thoughts that somewhere I am trying to link Symmetricity in the structure of Equations and the Graphical Plots. By Structural Symmetricity in the Equation of RZ, I tried to relate that to the game rules and also conservation of physicalities in the graph where 0 represents the singularity and physicalities is collinearity of non trivial zeros too. 

I then found that these counter examples have asymmetric aspects unlike Riemann zeta function in their structures and hence their game theory strategies would get distorted and hence also the graphical characteristics would likely be asymmetric and that’s why these counter examples might violate RH due to asymmetry and heterogenous in their structure of equations. 

On that basis, as per the Game theory rule or physical characteristics conservation, I tried to show that RH would be true for RZ function but may not be true for other Counter examples. There can be created many Counter examples but not as symmetric as RZ.

On that intuition and principle I tried to show RH is true for RZ ! I tried to conserve the symemtricity that were assumed while defining the arithmetic and algebraic rules of addition, multiplication, subtraction etc.. and establish the link between the symmetricity,homogeneity in algebraic equations and the graphical representation like for example the equation of circle is symmetric and homogeneous in algebraic structure and hence also symmetric and homogeneous in graphical representation.similarly for RZ function and Other Counter Examples satisfying the RZ functional Equations. I don’t see that an asymmetric equations will have graphical symmetry and symmetric equations curve will have asymmetric graphical structure !! That’s my main point.

So, I would kindly request to have some imaginative thoughts on my humble idea and reimagine Mathematics from a different angle. 

(I am trying to look from Natural Point of view not from Conventional approaches employed by expert number theorist. Because I see some possible fundamental limitations there . It’s like trying to lift the bucket by standing inside the bucket which won’t be possible !! 

This is because I believe that in mathematics one can prove many things but may not prove the axioms itself on which they have been defined. For that one has to possibly imagine the mathematical problem by treating it like a physical system externally as an observer not internally using arranging equations. 

Let's discuss mathematics unconventionally..

 

Riemann Hypothesis : Problem in Mathematics not of mathematics

Riemann Hypothesis: It's about Physics behind Mathematics. External analysis of Mathematics as the system . For example : Physics behind +,- etc..cant be explained using +,- itself ..Hence Trying to analysing  the mathematical tools  not possibly  possible using mathematical tool itself...

Riemann Hypothesis is about the Physics behind Mathematics itself...External Problem in Mathematics not Internal problem of mathematics that hence possibly not resolvable using the mathematical tools itself...For that one has to come outside the system.

Mathematicians are trying to lift the bucket up by standing inside the bucket..in context of Riemann Hypothesis.. Hence unable to do.

To resolve Riemann Hypothesis,one must look at mathematical system by standing beyond the system not inside the system .

That's why I had devised  an approach a decade ago to resolve the Riemann Hypothesis  earlier by being outside the system !! 

Need Higher Imagination in my humble view.

Science : Prediction without Predicting

In my experiential view, one should not think too much about time and outcome for effective time management ! Rather should focus on enjoying the processes locally in convex way, automatically things would be done over the time. Thinking too much would only make one fragile !!

The Best way to Optimize is Not to Optimize .

Best way to Predict is Not to Predict rather Focus on the Present Process Convexly..

Best way to achieve the Goal is to Focus on the Present...

As per Quantum World Science 

Universal Entanglement : Super Geometry of Quantum Entanglement

Brain internal dynamics is similar to the External Universe...Super Geometry of Quantum Entanglement 

There is Universal Interconnectedness Quantum Entanglement everywhere including Living Beings... Everything is Interconnected in the Universe at different levels. It's because of the Super Geometry 

Renormalization: Possible Incompatibility Application of Mathematical Number System to describe Physical realities !

Renormalization has been the most important fundamental issues in Physics. The Fundamental reason is incompatibility of the Number Systems and Mathematical Tools the way they have bee defined. there is fundamental mismatch between the intrinsic geometry between the mathematical number tools and the physical realities in Nature. 

It's like Reference frame where reference frame itself doesn't confotm to the Nature's Realities. 

This also leads to the possible  fact that Quantum Laws described in these Number system itself is Self-Inconsistent!!

Nature Realities  must be independent of Reference frames(Number system) .

There can be different Number Systems as reference frames but one muat check if their intrinsic characteristics are compatible with the Nature Realities or else they would lead to Contradictory results..that's the fundamental reason behind the disturbing infinities in context of Renormalization in Physics 

Science beyond Empiricism

Modern Science needs to go beyond Empiricism and Experimental Verification to broaden itself into True Science.

What if the moment an observer experiments, the outcome could vanish ! Quantum Effect !

So, taking Experiment as the base of Modern science could be narrow approach !!