The Towering Zeta Function

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 s i = + 1 2 σ . 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.


Introduction
The Zeta function seems to be the pyramid that holds the number systems together in a towering edifice of combinatorial relations.
Let  denote the complex numbers.They form a two-dimensional real vector space spanned by 1 and i where i is a fixed square root of −1, and , x y belong are real numbers, i.e.

{ }
: , x iy x y = + ∈  .The Riemann Zeta function is a complex variable function defined as ( ) σ > can be represented in terms of primes, p. ( ) s ζ is analytic for 1 σ > and satisfies in this half-plane the identity: ( ) Here, p is a prime.Except for a pole at 1 s = , ( ) s ζ behaves properly and can be easily extended using the Gamma function.The extension of ( ) s ζ to the entire complex plane can be obtained by consideration the entirety and the general definition of the Gamma function: ( ) Change variables by the substitution ( ) ( ) This can be split into two separate integrals, ( ) ( ) ( ) Note that the sum ( 6) is related to the Jacobi Theta function.See Ref. [1] ( ) ( ) ( ) The Jacobi theta function obeys the symmetry ( ) ( ) ) ( ) The integral (8.0) now becomes ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) The right side of the relation (15.0) is invariant to the substitution 1 s s → − .This gives the reflection for- mula for ( ) ( ) ( ) 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 s i e cos ln sin ln The ζ function can be written as a series in powers of n, where n is an integer, in the form: is when 3 n = , but has repeated values for 2 n = , at 4 n = , since ( ) From numerical calculations, the minima and maxima of the real and the complex parts of for 1 n > , occurs when n = 3.This is due to the fact that ln 2 ln 4 2 4 = . One sees that something special is happening between these three points, 2,   Again one sees that the minimum is at exp (1).
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.

Relationship of the Riemann Zeta Function to 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. [2] and others.

DEFINITION 1
Let n Z ∈ .A power tower is defined as a follows: Here, I am using the Knuth notation for the tower of powers raised n times.See Ref. [2].DEFINITION 2 ( ) I have used the case 0 n = as the argument itself, although in the literature most authors start with the defi- nition ( ) , 1.  define the future power tower as the iterates starting from the present value 1 n = as the first value that leads to a future th n value.Symbolically, ( ) ( ) ( ) ( ) 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

( )
W z solves for  the equation: W z is multi-valued and as has many branches with the usual notation for the th k branch as ( ) for the branch chosen.The principal branch of the function is and for real arguments, the function is denoted by ( ) W x .See Refs.[3]- [5].The branch points of ( ) , W k z take on real values only for These are subsets of the "Quadratrix of Hippias".Since  The fixed points of ( ) , log , log Obviously, z is constant over the range of values of c that satisfy the relation (31), thus the fixed points of the function ( ) converge to fixed points that satisfy (32) and exponential function of z, as will be seen later.For real values of x, the 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: ( ) ( ) Then the infinite number of possible inverse-solutions are paired as follows: Then these particular solutions of the function obeys the rules: , , then there exists a sequence of constants ( ) One can expand these functions as follows: As another general example, ( ) This can be converted to a product form: ( ) Here, 0 ∞   are independent integers.In general, ( ) Thus, taking Then, ( ) Thus,

( )
n z ζ  is a power tower that can be put in the general form: This however is only valid for ( ) 1 z ℜ > .Now consider the same power towers in terms of primes.In terms of primes, let 0 b p be the 1 1 th b = , prime with 1 2 p = .Then, ( ) ( ) The Zeta power tower in terms of primes becomes the simple form: 3. Convergence of ( ) Consider the extended Zeta function, (1 ) (1 ) ( ) ( ) ( ) The iterated zeta function becomes: ( ) 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 can be separated into the product of two power towers: e 2 2 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 ( ) converges to a constant whenever there exists some root of the Towering Zeta function,  that satisfies . Thus, all the arguments that lead to a root are unique and no periods of the function can exist in the past iterations.This does not exclude close to periodic regions.This subject has been extensively studied in Julia Sets theory.
This can be seen when the function is taken to the limit, and becomes an image of itself.
( ) ( ) ( ) The solutions to the relation ( ) converges for real values when n is large and obviously the power tower product converges to the constant function, If there appears an argument is the fixed point and the real limit of the function ( ) s is a root of the Zeta function.The convergence becomes real and the complex part of the arguments vanish.A plot of these values for ( ) is shown in Figure 4 for the converging values.Thus c ∞ is a super attractor for all roots of the Zeta function.
Any roots r s of the Zeta function that appears in any iteration of the function at the th r term, r n < , will converge the function, ( ) The functions of the infinite iterative form: have certain attracting and repelling values, and are sometimes periodic with respect to n.If one takes roots r s of the function as a starting point one finds that the function generates constants for each value of n, such that: Then, for all roots r s of the zeta function, there exists a unique sequence of real constants that are invariant with respect to all the roots and that converge to the same value, c ∞ .The plots below show examples of how the function's values change for the real arguments generated by some complex roots.

τ =
One can see that the function has alternating maxima and minima at integer values of n with the lowest value occurring again at 3 n = .This is again due to that fact that the function converges to a constant for all roots as n → ∞ .
One can surmise that any root r s such that ( ) r s ζ vanishes will generate a convergent sequence of real arguments k s , for the functions 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 r s , one could replace zero of ( ) 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.

Convergence of the Riemann-Zeta Function for Complex Values
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 z can be classified depending on the value of the derivatives, ( ) . Such points are studied in Fractals, Chaos theory, Attracting Periodic Cycles, and in Mandelbrot Sets, using Newton approximations and this critical point is classified as

Neutral if 1
Repelling if Then, the function , if and only if m divides p.This product is referred to as the multiplier of a periodic point z of period p.Since the i z lie on a cycle, ( ) ( ) ( ) ( ) 0 for all .
if and only if m divides p q − .LEMMA 2: If s is a root of the Zeta function, then the Towering Zeta function, ( ) ( ) , where r is a positive integer.Then there exists some ( ) ( ) i.e. there exists a sequence of arguments, ( The iterated exponential that generates a root ( ) , must satisfy all roots independently, since if for some r, r z is a root of ( ) z must generate a particular sequence of real arguments in a future se- quence of arguments that must converge to c ∞ , as k → ∞ , i.e., ( ) ( )  Such a sequence of real arguments cannot generate another complex root in the future direction.As an aside, Little wood showed that if the sequence , 1, , k z k r =  , contains all the imaginary parts of all zeros in the upper half-plane in ascending order, then, There cannot exist such a sequence of roots in the past or future arguments of ( ) since when a root is encountered the sequence of arguments converge and never goes to zero but once.However, considering the fact that the Zeta function is multivalued at the roots, any root could be used in the future of a Zero, and as such a product of all roots following the Hadamard product for the Zeta function can be used in the future of a Zero of the function.
The Towering Zeta function

( )
n s ζ  can in fact have an infinite number of convergence points in all its roots.
All roots converge the function to the real line from the complex plane (Figure 5).
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.Table 1 shows the iterations from a root s in an upward future trend toward convergence to c ∞ .However starting from some past iteration, there exist an infinite number of roots that could be generated by a past iteration through zero.
Table 1 shows successive past values of arguments for the roots, that solve the iterated relation, ( ) ( ) , where n runs up the tables in a future direction, i.e. arguments of ( ) As can be seen from ε cannot be zero.However, we can choose 0 z and make ( ) ( ) Then there exists a negative constant such that ( ) The constants 0 t must vanish at infinity since ( ) , and any approximations of a constant function must be a constant-function, then, one suspiciously finds that: This value deviates from π by 0.003787727780215700 .I will now discuss the relationship between the Towering Zeta function ( ) Thus the derivative of the function is just products of the derivatives of the iterate functions taken over 1 n − values.Note that the derivatives of ( ) have been shown by D.L. Shell [6], to be periodic.Obviously, the power towers are intimately related to the Towering Zeta function and the almost periodic relationship is exactly the sort of behavior one sees with the Zeta function: LEMMA 4: Which is the same as the power tower derivatives when c e = . ( Noting that ( ) The derivative becomes, ( ) Put 1 k a u + → , and use ( ) ( ) ( ) Consider the product formula: Let the root s occur at the th m iteration of the product formula, at k m = .Then, ( This separates the products into three terms, the first term, P, being the iterates before the root s is encountered, 0, , and the second term N, being the iterate that produces a root at N k m ∈ = , and third term being the iterations after the root at 1, , 1 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. , and if , , , Z p q m n ∈ , and if 0 p q m n < < < < , and ( ) ( ) , then, ( ) . Then, iterations from p to q will be periodic and will only generate cyclic arguments when , where, k q p = − .Thus the past iterations for i q < , will be stuck in an eternal loop and never generate future roots, s.This is true for both real and complex arguments, z and for real and complex roots s.
LEMMA7: Let ( ) p q m k ∈ , and 0 p q m < < < , and z ∈  , and if for any p, q, ( ) ( ) , then, P and F are infinite power towers.Proof: If , , , Z p q m n ∈ , and 0 p q m n < < < < , and if for any p or q, ( ) ( ) then the sequence of iterations of the zeta function ( ) has no past purely periodic arguments and as such there can be no other repelling, neutral or super-attracting points until the root itself is reached.
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 ( ) The same argument leads F to an infinite power tower.Noting that ( ) 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 ( ) , then, the derivatives of ( ) must vanish since it converges to a constant.Now the conditions that allow a root to be encountered depend on P.
It is worth noting that the function , and the arguments ( )


, are discontinuous functions over the complex plane.In other words they jump in values over each iteration and may never hit a root for some values of z.
Let 0 z be a starting value that does not hit a root.The relationship given by the Hadamard factorization thus represents the relationship between the iterated functions ( ) , and roots ρ of the zeta function.
Obviously, the argument ( ) , is a pole of the function and so we assume ( ) From the reflection formula, ( ( )  never hits a root, we can divide across by the reflection function, , one can divide across by ( ) ( ) ( ) From the reflection formula, ( ( ) ( ) From the power of π , the condition for no roots to be obtained during iterations is that ( ) ( ) ( ) ( ) , the iterates that lead to a root give From, (124), one sees that the derivative vanishes at the root of given by some iterate ( ) . The condition that the derivative does not vanish when , leads to a solution of the Riemann Hypothesis.However, for the case is a root, none of the iterates vanish, since the iterates lead to the final root.

Connection of the Towering Zeta Function to Exponential Power Towers
To illustrate the convergence of the Towering Zeta function for ( ) , we start with: Then, ( ) To see that the reflection formula obeys the power tower representation, take, Then, since , the two zeta functions can be written in the form: LEMMA 8: The fixed points of the Towering Zeta function are rational functions of its roots.Proof: The fixed points are obtained by the solutions to the relation: This can be simplified by putting ϕ are the same.Thus for the complex values, if there is an infinite number of unique divisors, then, ( ) ( ) ϕ must be divisible by n φ leaving a factors that are the roots of unity.To see that the remaining factors are the roots of unity take , and so, and from this Write the rational function ( ) , and, Thus the reflection formula applies to the power tower representation: The reflection formula then tells us that where n φ  denotes the th n iterate of the map φ and converges.
An even stronger condition can be placed on the convergence of the sequence (143) by Thron in Ref. [8].LEMMA 10: (Thron; Ref. [8]): If e x x a = , 1 x < , or if x is a root of unity, then the sequence (143) converges to e x .For almost all x such that 1 x = , the sequence diverges.
Galidakis Ref. [9] noted that the fixed points ( ) .I.N.Baker and P.J. Rippon Ref. [10] showed that if θ is a centrum number, in particular if θ is non-Liouville number, then (143) diverges.The th n order power tower only converges to n fixed-points on the circle when ( ) , a root of the function and when Following Titshmarch Ref. [11] page 14, taking

R
However the iterates are associative, thus, ( ) ( ) Of course, this is only true if the backward iterations are uniquely selected from the initial argument set ( ) . The range of the function changes since, ( ) ( ) k-iteration of the function for arguments in the range ( ) ( ) , give the functional relation. ( In other words, after convergence, the relation obeys the functional, ( The relation gives the invariant integral over an infinite iteration of the roots: I now introduce the function ( ) x ϕ defined by the following theorem due to Ramanujan.Chakravarthi Padmanabhan Ramanujan was born in India on the 9 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.LEMMA 12: There exists exponential power towers such that if The following Theorem by Thron, [8], applies to such power towers.
For almost all ( ) LEMMA 14: For any complex numbers, , x LEMMA 15: For any complex numbers, x, + , then, the self-root function is given by: (

∑ ∑
Proof: In [12], and in [13], Jovovic calculated the self-root sum: where is the Sterling number of the first kind.Thus, for the self-root, Note that in general, one can write: In which case the function  ( ) Differentiating ( ) ( ) , k times, a simple calculation shows that: where the symbol ( ) is the value of the th k derivative at 0 x = .This is the same expression as ( ) ) ( ) Now from Euler-Mac Laurin summation formula: Noting from the Bernoulli relation that only odd values of k survive, the integral (171) is zero when Consider the Fourieh series expansion, ( ) ( ) The derivatives ( ) ( ) ( ) ( ) ( ) ( ) Thus, the fractional functions ( )

( )
F x satisfy Ramanujan's Master Theorem, when ( ) ( ) ( ) , and so and s ψ is not a function of z, then the Ramanujan function is related to the Riesz function.

Proof:
Taken over square-free numbers m, and squarefull numbers n, then: The number of both the squarefree and the square-full numbers is given by ( ) this is the one by one count of each infinite set.Thus, the density of the squarefree numbers and the squarefull numbers is given by: One sees that the paired functions ( ) ( ) ψ satisfy the Master Theorem.Thus, from LEMMA 25 ( ) Further, to see that this is true, define:  f x that has derivatives of all orders throughout a neighborhood of a point ξ , may be written as:

Relation of Power Towers to Transcendental Numbers
As can be seen from Figure 6, the function  LEMMA 27: (Mladen, [14]) For every real number 1,    g z had an infinity of roots of unity, it would have elements 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 n ϕ is divisible by all the factors of n φ then, the power towers can be written as 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 e , 0, 0 iax x ia − ≥ < and thus will always contain a con- tinuous banded group character if the function does not vanish identically.

( )
f s belong to a space ,1 Lp p < < ∞ , and let it not vanish almost everywhere on any interval 0 s a < ≤ .Then Lp contains at least one exponential function of the form e λ .
The function ( ) s ζ can also be represented by exponential terms and, rational functions of its roots can also be represented by the exponential forms.Since the function is analytic everywhere except for a simple pole with residue 1 at Any form of ( ) s ζ that expresses its roots as an entire function can be used to relate to exponential forms.There are many forms of the function that can be represented as rational functions of the roots when it vanishes.The exponential function is represented by the quotient of conjugate rational complex functions on the half-line: 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: , then only the fractional parts of the arguments contribute to solution of (232) and thus all the complex roots of the function will be on the 1/2-line.
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:      With the general condition that 2 2 π π α − < < , these solutions satisfy only the real negative even roots of the 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.

Discussion of the Result
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.Table 3 shows the almost periodic cycles of the function that appears to dominate some roots.The inverse iterations seem to generate "very near root" misses.
gives the relation: also write the Zeta function as follows:

Figure 2 ,Figure 1 .
Figure 1.Shows that the minimum values of

Figure 2 .
Figure 2. Shows the maximum value of

Figure 3 .
Figure 3. Shows the graphs of the real component


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.
the real values of the function occur at the branch points ( )

DEFINITION 7 :
is defined as a future iterated exponential by taking the ζ-value of (z) and then taking the ζ-value of process n-times.The arrow shows the direction of iteration of Towering Zeta functions and the arrow → means take increasing nests of ζ values of prior Zeta values to obtain a new future value.Define the inverse past iterated exponential of the Towering Zeta function ( ) ( )

1 ζ−
where the inverse-zeta function is one of the set of infinite solutions to the equation,( ) of the Zeta function.Then, the future iterates ( ),1, , the function is continued over values of k for a given root, r s ,

Figure 4 .
Figure 4. Shows the oscillations for convergence of iterates from a root.

Figure 5 .
Figure 5. Shows the convergence from positive complex roots of the function to the real line.

,
be a non-vanishing function on the disc 0 s r ≤ , which is analytic in the interior.Then,

(
the condition demonstrates that any prior argument that is a root another root again.

) 1 .
Power towers of the form (145) represent rational functions.Remark 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:

8 .
. In Ref.[5] Shkliarski D., N. Chentzov, and I. Yaglom, show that for real integer values of s, the forms of equation (142) represent Power Towers that converge to real rational values.DEFINITION Define the set

nw
converges with limit λ ∈  , then ( ) λ converges to e x , in some neigh- borhood of e x , if 1 x < and can do so only if 1 x < .Proof: Take the principal branch 0 k = .
θ being a rational function.Following Titshmarch, Ref. [11], Let ( ) x φ be any function with a continuous derivative in the interval { } , a b .Then, if x     denotes the greatest integer not exceeding x, ( to the Ramanujan Master Theorem and to the convergence of the Power Tower Zeta function.

DEFINITION 9 :=
For any real or complex numbers, , Then using the Master Theorem, the following apply:

LEMMA 21 :
For complex values of z, if ( )

192) 6 .
Relationship of the Function( ) F ψ α to the Density of Squarefree and Squarefull (Non-Squarefree) Numbers DEFINITION 10: A positive integer m is squarefree if it is either a product of different primes or 1 otherwise it is squarefull.Let of number of squarefree numbers and squarefull numbers in an infinite set of integers.Note: RIEMANN'S HYPTOTHESIS: Fix 0 >  .Then we can find N such that for all n N > the number of square free numbers in [ ] 1, n does not differ from the number of non-square numbers in [ ] The density of squarefree numbers is ( )

203) LEMMA 26 :
If a function, ( ) f x , has derivatives of all orders throughout a neighborhood of a point ξ, then Ramanujan's Master Theorem is simply the Taylor series.Proof: The Taylor series of any function ( )

1 xx
only has quadratically equal values (red x's) at 2, 4, x = and also at 1, .x = ∞ There are no other paired quadratically equal values of  that have the same values for 1 xx Hence the following LEMMA due to Mladen[14].

Figure 6 .
Figure 6.Shows a plot of the function f x is a continuous function strictly increasing on the interval

1 x
> be an algebraic number such that ( ) ( ) that ( ) 1 g z = , and the values of x are just the products of the ( ) with k.If ( ) are entire functions.An Entire Functions has special properties that relate to exponential functions.If ( ) f z is analytic for the entire complex plane, then it is an entire function.An entire function can be represented by in the linear form:

1 s
entire function, and so one suspects that when the function vanishes, there exists an exponential representation of its roots i the complex roots obey the exponential relation (232) for the arguments,

Figure 7 .
Figure 7. Values of iterations of the root s = −2.

Figure 8 .
Figure 8. Values of iterations of the root s = −4.

Figure 9 .
Figure 9. Values of iterations of the root s = −6.

Figure 10 .
Figure 10.Values of iterations of the root s = −8.

Figure 11 .
Figure 11.Shows the plots of inverse iterates of real negative roots.

Table 1 ,
the function can become almost oscillatory for some values of negative roots.Values down the table are arguments that can generate roots as starting arguments of successive values of

Table 1 .
Iterations of the roots of the Towering Zeta Function over real arguments.

Table 3 .
Iterations of the arguments below seem to generatealmost periodic cycles of the set below.