_{1}

^{*}

Over a century and half has passed when Bernhard Riemann hypothesized that the non-trivial roots of the Riemann zeta function ζ( s) all lie on the half-line . In this paper the Zeta function is iterated as a power tower and its properties are applied as an approach to an indication that the Riemann hypothesis might be true. It is known that complex valued Power towers converge under certain conditions to exponential power towers of entire functions. These properties can be used to resolve the Riemann Hypothesis.

The Zeta function seems to be the pyramid that holds the number systems together in a towering edifice of combinatorial relations.

Let

Here, p is a prime. Except for a pole at

Change variables by the substitution

Extracting

The convergence of the series,

in the interval

This can be split into two separate integrals,

Note that the sum (6) is related to the Jacobi Theta function. See Ref. [

The Jacobi theta function obeys the symmetry

Thus

The integral (8.0) now becomes

The right side of the relation (15.0) is invariant to the substitution

The reflection formula indicates that the roots should obey a reflection and conjugate symmetry if they lie on the 1/2-line.

One can also study the maxima and minima of infinite products powers by looking at the functions that approximate the ζ-function.

Proposition 2: The Zeta function is related to power towers.

Let

The

where,

Note that in (21), as

is when

tion as follows:

From numerical calculations, the minima and maxima of the real and the complex parts of

between these three points,

One can see that all three functions are somewhat related by the relation s:

This is true for the functions,

One finds that these functions have their maxima and their minima in the range,

This is exactly the range where real power towers converge. I will use this later when I discuss power towers.

Power towers have been studied extensively. I start by describing power towers following some conventional methods that have been used by Knuth Ref. [

DEFINITION 1

Let

Here, I am using the Knuth notation for the tower of powers raised n times. See Ref. [

DEFINITION 2

I have used the case

It is understood that such a power tower is iterated from some past argument to its present argument. By past I mean, the values that would have occurred in an iteration of the function that lead the iteration to its present value.

DEFINITION 3. If the limit of

DEFINITION 4. For

It is understood that such a power tower is iterated from its present state to some future value. By future I mean the argument values that will occur in an iteration of the function from its present value to some future value.

DEFINITION 5. The complex Lambert W function

other branch point that has real values is

These are subsets of the “Quadratrix of Hippias”.

Since

Consider the function,

Let

Proof:

Using the Lambert W-function,

Obviously, z is constant over the range of values of c that satisfy the relation (31), thus the fixed points of the function

If

function of z, as will be seen later. For real values of x, the function

How does this relate to the iterations of the Zeta function?

DEFINITION 6: Define an iterated exponential as the Towering Zeta function:

Here,

DEFINITION 7: Define the inverse past iterated exponential of the Towering Zeta function

Here the arrow ← means take decreasing past nests of

Since the solutions to (36) are multivalued, we wish to fix particular solutions that will revert the function values back to original values from its inverse values.

Start with the reflection formula:

where,

Let there be solutions

Then the infinite number of possible inverse-solutions are paired as follows:

Then these particular solutions of the function obeys the rules:

For example, if only consider

One can expand these functions as follows:

As another general example,

This can be converted to a product form:

Here,

Thus, taking

Then,

Thus,

This however is only valid for

Now consider the same power towers in terms of primes.

In terms of primes, let

Thus, taking

The Zeta power tower in terms of primes becomes the simple form:

Consider the extended Zeta function,

Then,

Using this,

Further,

Thus, taking,

The iterated zeta function becomes:

Note that

This is just the relation for the general zeta function:

One sees that this is also a power tower product but this time the sum and the product operators are doubled. Thus the iterated exponential power tower of the zeta function for

Since the raising power is not over the entire product, this becomes difficult to write in the general Knuth form. The future iterates of the function

Let

Obviously, the function vanishes for

When the function is continued over values of k for a given root,

This can be seen when the function is taken to the limit, and becomes an image of itself.

The solutions to the relation

Thus

If there appears an argument

Any roots

have certain attracting and repelling values, and are sometimes periodic with respect to n. If one takes roots

Then, for all roots

One can surmise that any root

Assuming the Riemann Hypothesis, there is a symmetry between the real parts of the known complex roots s, of the function about zero:

This symmetry is due to the fact that at any point in a large number of iterations where a root

This implies that the convergence of the iterations of functions must be symmetric about a zero of the function as implied by the mean value theorem mention earlier.

It is worth noting that the complex parts of iterates of the complex arguments result in real arguments if the products of the components of the iterates is real. Thus the roots are expected to be composed of a spectrum of complex factors whose iterates are real since they result in a quadratic convergence to the real values due to the symmetry of the reflection formula.

Let Z belong to integers and Z^{+} to the subset of positive integers. Any sequence of arguments can be created by functional iteration. Let function

Then, the function

・ If

・ If

LEMMA 2: If s is a root of the Zeta function, then the Towering Zeta function,

Proof: The case

i.e. there exists a sequence of arguments,

Then,

The iterated exponential that generates a root

There cannot exist such a sequence of roots in the past or future arguments of

The Towering Zeta function

Starting from a given root, the future iterations of the Towering Zeta function function over successive arguments that start from a root will lead to a convergence for every root.

: | : |

17 | −0.29595806723778959429546880727279905000000000000000 |

16 | −0.29580152831200018096290646344253482702467849073213 |

15 | −0.29610685544242180860201784355977774292750241992426 |

14 | −0.29551147448528153468605540107430168095507096662307 |

13 | −0.29667304591820569539524370288435971088621053886270 |

12 | −0.29440910387263694790663240676204680320270972628433 |

11 | −0.29883017491388564173005702091107050051340373827686 |

10 | −0.29022915571595152652543096855529715421054509417251 |

9 | −0.30708636451194022129234803037380574262347512272602 |

8 | −0.27451684815677210287939135693311577140783360600261 |

7 | −0.33925706658308498350126705482505739973880105347080 |

6 | 0.21728231379886310314230045114591184991388919709022 |

5 | −0.47450768974007172623842641724875639785397472932299 |

4 | −0.02860979985485943088252867867522696522057632451858 |

3 | −1.4603545088095868128894991525148973697655036386338 |

2 | −1/2 |

1 | 0 |

s |

LEMMA 3: Le t

Use Voronin’s theorem and let

Further, since

So that

As

Then there exists a negative constant such that

Then,

The constants

This value deviates from

I will now discuss the relationship between the Towering Zeta function

Thus the derivative of the function

have been shown by D.L. Shell [

LEMMA 4:

For

Proof:

Differentiating

Which is the same as the power tower derivatives when

Hence,

Noting that

The derivative becomes,

Put

LEMMA5: Define

Proof:

First we verify (102).

Consider the product formula:

Let the root s occur at the

This separates the products into three terms, the first term, P, being the iterates before the root s is encountered,

One has to determine if the factors P, N, F, can vanish in a given range of values of the arguments, s. Before determining these products, the following Lemmas are necessary.

LEMMA 6: If

Proof: Suppose there exists

LEMMA7: Let

Proof:

If

Then by induction, there are no past periodic arguments in the factor P prior to encountering this root. There must exists an infinite past for the iterations leading to root s. Then, the only critical point is the root itself and the sum of all such points will be the root. The root could be written as an infinite power tower of

Noting that

Thus

The factor N is given by:

Both the above factors are zero, hence the proof. It is obvious that if a root occurs in any of the arguments of the function iterates of

It is worth noting that the function

Let

The relationship given by the Hadamard factorization thus represents the relationship between the iterated functions

Obviously, the argument

Since

Since

Using

From the reflection formula,

From the power of

The condition

tion. Thus the condition demonstrates that any prior argument that is a root

Further, for

From, (124), one sees that the derivative vanishes at the root of given by some iterate

tion that the derivative does not vanish when

Hypothesis. However, for the case

To illustrate the convergence of the Towering Zeta function for

Then,

Now use

To see that the reflection formula obeys the power tower representation, take,

Then, since

LEMMA 8: The fixed points of the Towering Zeta function are rational functions of its roots.

Proof:

Putting

Thus,

And so,

The fixed points are obtained by the solutions to the relation:

This can be simplified by putting

Power towers of the form (145) represent rational functions.

Remark 1. The only real algebraic solutions to (137) are 1, 2, and 4, since as shown in (23), functions of the power form (137) have an equivalence for the values 2, and 4:

It is easy to see that if

For real power towers, the prime divisors

and from this

Write the rational function

Then,

Thus the reflection formula applies to the power tower representation:

The reflection formula then tells us that

DEFINITION 8. Define the set

where

and so on with each such relation representing an entire function of

LEMMA 9: (Shell: Ref. [

Proof: Take the principal branch

An even stronger condition can be placed on the convergence of the sequence (143) by Thron in Ref. [

LEMMA 10: (Thron; Ref. [

If

Galidakis Ref. [

Following Titshmarch, Ref. [

Let

Following Titshmarch Ref. [

Since

For the region

and so

Assuming the Riemann Hypothesis, the Towering Zeta Function follows this relation before convergence, i.e. when the real part of the arguments of the Zeta function is in the range

Let s be a root of the Zeta function. Assuming the Riemann hypothesis, when the Towering Zeta function converges, the next argument is of the form

Thus, for some finite iterations of a starting argument, z, let

However the iterates are associative, thus,

Of course, this is only true if the backward iterations are uniquely selected from the initial argument set

k-iteration of the function for arguments in the range

In other words, after convergence, the relation obeys the functional,

Then, for

The relation gives the invariant integral over an infinite iteration of the roots:

I now introduce the function ^{th} of January 1838. He died on the 27^{th} of October 1874. He has been referred to as the greatest mathematician ever. His work on number theory and algebraic geometry has produced some of the most outstanding revelations in mathematics and is considered to be one of the pillars of modern day research. This paper is about Ramanujan’s so called Master Theorem that relates integrals of certain types of functions to a wide range of application including Power towers and the Zeta function.

The function

LEMMA 11: [Ramanujan’s Master theorem Ref. [

Define

DEFINITION 9: For any real or complex numbers,

Then using the Master Theorem, the following apply:

LEMMA 12: There exists exponential power towers such that if

Proof:

LEMMA 13 {Thron}: If

LEMMA 14: For any complex numbers,

Proof:

LEMMA 15: For any complex numbers, x, if

where,

Proof: In [

where

Note that in general, one can write:

In which case the function

LEMMA 16: If

LEMMA 17: For the self-root if

Proof:

Differentiating

where the symbol

LEMMA 18: If

Proof: Using the Master theorem,

Now,

From the Zeta function, since

Now from Euler-Mac Laurin summation formula:

Putting

Noting from the Bernoulli relation that only odd values of k survive, the integral (171) is zero when

LEMMA 19: Define

Proof:

Consider the Fourieh series expansion,

The derivatives

Put:

then,

Thus, the fractional functions

Then,

^{*}Note that putting

Then, as

LEMMA 20: If

Proof:

LEMMA 21: For complex values of z, if

Proof:

LEMMA 22: For complex values of z, if

is related to the Riesz function.

Proof: Integrating (181), and since

Note that (182) is exactly the Riesz function for

LEMMA 23: For

Proof:

If

Using the Zeta functional relation,

Relation (185) becomes:

For

Corollary: If

Proof:

Thus

It is worth noting that relation (190) is one form of the Weyl fractional derivative. See Ref. [

DEFINITION 10: A positive integer m is squarefree if it is either a product of different primes or 1 otherwise it is squarefull.

Let

Note: RIEMANN’S HYPTOTHESIS: Fix

LEMMA 24: The density of squarefree numbers is

Proof:

Taken over square-free numbers m, and squarefull numbers n, then:

Since

Thus,

Let

Then,

The number of both the squarefree and the square-full numbers is given by

thus,

It follows that the density of square-free numbers is

LEMMA 25: Let

Proof:

Taking the derivatives,

One sees that the paired functions

Further, to see that this is true, define:

Put

LEMMA 26: If a function,

Proof: The Taylor series of any function

Thus as

As can be seen from

LEMMA 27:

(Mladen, [

a)

b)

c)

and for

LEMMA 28: Let

a) For

has exactly two solutions,

b) For

c) For

LEMMA 29: Let

a) if

b) if

and the infinite Power Tower

c) If

then the infinite Power tower

Proof of Theorem 1: For

Let

Starting with the Zeta function for

Let

Taking

The sequence of integers

The sequence of integers

Factoring;

By the Lindemann-Weierstrass theorem,

is algebraic and the product of a transcendental number

Thus we are left with the relationship:

where, A is an algebraic number. See Ref. [

The only value for which the left hand side and the right hand sides are transcendental is when

The Power Tower of the pure complex form with

not only follow Ramanujan’s Master Theorem, it also converge to exponential functions of the form:

It is clear that the function converges only when

This just shows the self-similarity of the convergence.

Now the only arguments that satisfy the path to convergence are:

See Ref. [

There are an infinite number of possible roots that can independently satisfy (218). For example, if a root is obtained from the backward iteration,

Thus any complex root s can be obtained by iterations that go backwards from the real negative line provided the arguments are chosen to be of the rational form:

The solutions to inverse Zeta values is referred to as “a-points”. These points have been extensively studied by other authors, see [

also require that

The roots of the zeta function obey this condition as can be seen from

Remark 2: The arguments that lead to roots have singular solutions to the inverse zeta functions of roots that generate them in the backward direction. This points to the fact that a root can only be arrived at when there are no cycles prior to the root. Thus, any argument z that leads to a root can be backward generated for as singular values of the inverse zeta function that it generates. Not all values of z can generate a root. Those that generate a root must conform to the power tower structure. The only reciprocal relations that relate the symmetry is

From this, one could surmise that the reflection formula and the inverse zeta power towers prescribe conjugate power towers that are also reflection symmetries about unity for the Zeta function. Write the Zeta Power Tower as follows:

S | |
---|---|

0.07071826295 | |

0.04756015651 | |

0.03997731031 | |

0.01011817652 | |

0.00 |

It is clear that

The degree of

of arbitrarily high degree, and thus would not be of finite degree over the rationals, and thus would not, in fact, be an algebraic number field. Thus, equating the powers of (223), if

These Power Towers are entire functions. An Entire Functions has special properties that relate to exponential functions. If

Anne Beurling demonstrated in all cases the translates of an integrable function defined in the entire interval [0, ∞], are represented by at least one exponential form

The function

is an entire function, and so one suspects that when the function vanishes, there exists an exponential representation of its roots

Any form of

In all these instances one finds that if the roots are on the 1/2-line, and

with

For the real arguments one finds that if the roots are on the 1/2-line then they must obey a certain symmetry that satisfies:

If the complex roots obey the exponential relation (232) for the arguments,

For the pure complex arguments, the roots will not satisfy the symmetry (223) since

Exponential-functions with pure complex argument will not be found since the right-hand side is real while the left-hand side is pure complex. For solutions with arguments that are complex, if the roots are on the half line, they must satisfy the symmetry:

−0.955672796977804042554378959475171320670 |
---|

−30.0000000063544093856041366895197085049 |

−41.9999999999999998574302112018735692239 |

−0.976512314920661294744350276007417719132 |

−22.0006303025081290011252788909666361481 |

−21.9855218713537597568162560452633382547 |

−21.9855320025224369786834301184732282964 |

With the general condition that

function which are the only known real roots of the function. Thus if the roots are on the half-line, the only exponential arguments that will satisfy the roots are for the complex conjugate roots on the half-line. This indicates that the arguments that yield solutions to the vanishing of the Riemann-zeta function are symmetries that satisfy inverse tangent relations and as I will demonstrate in future papers that the arctangent symmetry (223) relates the Bernoulli numbers, Zeta functions, and the Gamma functions to prime numbers.

The convergence of Power towers relates the vanishing of the zeta function to the half-line. This relationship comes from the property of complex power towers of the exponential-form only converge to exponential functions relating the roots to the convergence. If one iterates backwards from a real root, one finds near misses of purely periodic states of the function as shown in Figures 7-11. Obviously if the cycle ever gets to be purely periodic then no roots can be generated since the periodic cycle will prevent any root from being generated backwards from the infinite past.

Michael M. Anthony, (2016) The Towering Zeta Function. Advances in Pure Mathematics,06,351-392. doi: 10.4236/apm.2016.65026