Approach to Riemann Hypothesis by Combined Commensurable Step Function Approximation with Bonnet Method

To the Riemann hypothesis, we investigate first the approximation by stepwise Omega functions ( ) u Ω with commensurable step lengths 0 u concerning their zeros in corresponding Xi functions ( ) z Ξ . They are periodically on the y-axis with period proportional to inverse step length 0 u . It is found that they possess additional zeros off the imaginary y-axis and addi-tionally on this axis and vanish in the limiting case 0 0 u → in complex infinity. There remain then only the “genuine” zeros for Xi functions to continuous Omega functions which we call “analytic zeros” and which lie on the imaginary axis. After a short repetition of the Second mean-value (or Bonnet) approach to the problem and the derivation of operational identities for Trigonometric functions we give in Section 8 a proof for the position of these genuine “analytic” zeros on the imaginary axis by construction of a contradiction for the case off the imaginary axis. In Section 10, we show by a few examples that monotonically decreasing of the Omega functions is only a sufficient condition for the mentioned property of the positions of zeros on the imaginary axis but not a necessary one.


Introduction
In his article [1]  ) that remained unproved up to now. This function was already known to Euler (Euler product) and using the uniqueness of the prime-number decompositions of natural numbers 1, 2,3, n =  Euler established in about 1737 the connection to the sum form. Its main importance is as Riemann showed in [1] that one can derive from it approximations for the prime-number distribution. Many articles and monographs are published since this time to this function, e.g., [2]- [14] (some important original articles of the past are republished plus a few expert witnesses are published in [3]) 1 .
Concerning the trivial zeros of ( ) Then the conjectured zeros lie directly on the imaginary axis. There are many known functions with zeros only on the imaginary axis, in particular, the entire modified Bessel functions Our first and main source for detailed serious studies in this field was the book of Edwards [2] after knowing the problem much earlier mainly from popular articles and books to recreation mathematics. where we applied partial integration. With separation of the real and imaginary part we find as is was given in similar form in [7] (chap. 17.7, Equation (12) and Equation (14)) and in [2] and as it was also derived in [15]. For the modified entire Bessel where ( ) although, astonishingly, this is not immediately to see from its given explicit form (see remark in [15]) and also the form (1.11) for the Omega functions to modified Bessel functions possesses this symmetry. Besides to be functions which rapidly decrease in a way that the integrals of the form (1.6) exist the Omega functions (1.10) and (1.11) are monotonically decreasing for 0 u ≥ and it was the conjecture in [15] that this is a common  [17] (chap. 5, §2), Widder [18] (chap. 5)) can be applied which provides a function which depends on the initial and final values of the Omega function and which possesses in the argument a mean-value function with some known properties originating from the applied averaging process. For (1.6) assuming analyticity it possesses the form where 0 Ω denotes the lowest moment of the function ( ) u Ω which does not depend on the reference point.
In our article [15], apparently, this was considered correctly only up to discussion of the zeros on the imaginary axis in case of continuous Omega functions and considering the principal form of the functions for zeros on axes parallel to the imaginary axis. The last was made by actions onto the function on the imaginary axis by operators. From the correction to [15] it seems to remain correct the statement that only the monotonically decreasing step-wise constant functions with equal step-lengths have to be excluded with additional zeros. The article [19] was mainly intended to illustrate the behavior of zeros when going from lower to higher Taylor-series approximations but to the end of its elaboration it became almost evident that considering step-wise approximations of the Omega functions with commensurable step lengths 2 and finally going with the step-lengths to zero is an appropriate method for considering the problem where temporal zeros in the approximations off the imaginary axis and on this axis in the limiting process go to complex infinity and there remain only the genuine zeros. To carry out this systematically is a main aim of the present article.
Since all nontrivial zeros of the Riemann zeta function can never be found explicitly this suggested from the beginning of the work that we have to look for methods which do not need the exact zeros and rest only on general properties of the considered functions and which, therefore, are true for a whole category of functions including the mentioned ones. We find that in the step-wise approach with commensurable step lengths there exist two different and well-separated kinds of zeros on the imaginary axis where the first kind possesses also zeros off the imaginary axis and in the limiting case of vanishing step lengths go to complex infinity whereas the second kind stabilizes in the limiting procedure of vanishing step lengths on the genuine zeros of the Xi functions to the considered Omega functions and, apparently, do not possess zeros off the imaginary axis (Sections 3 -5 and also 8). This is underlined via the Second mean-value approach or Bonnet method of integration which can be successfully completed and since in this method the transition from the imaginary axis to axes parallel to the imaginary axis is possible using the Cauchy-Riemann equations in an in- 2 To this case belong also functions with only rational proportions of step-length (that means commensurable step lengths) for which then the smallest common part of all step lengths can be taken as (genuine) step length u0. tegrated form. We repeat this approach in short form in the Sections 6 -8 and give in Section 8, apparently relatively simple, proof that the "genuine" zeros in continuous case lie only on the imaginary axis. In Section 10, we show by example that the (strictly and non-strictly) monotonically decreasing of Omega functions in the limiting case of vanishing step lengths is a sufficient criterium for zeros only on the imaginary axis but is not a necessary one. An advantage of the mentioned methods is also that they provide arguments that step-wise constant Omega functions with incommensurable step-lengths (plus monotonically decreasing) fall also under the category for which the Xi functions possess only zeros on the imaginary axis.

Commensurable Step-Wise Approximations of Omega Functions to Riemann Xi Function and to Other Appropriate Xi Functions
We approximate the (in general, non-strictly) rapidly decreasing function by a step-wise constant function with equal step lengths 0 0 u u ≡ ∆ > (a parameter which finally in a limiting procedure we let go to zero) of the following form ( Figure 1) where ( ) where we introduced the abbreviations ( ) ( ) as monotonically decreasing functions according to (2.1) they have to satisfy the inequalities  Finally we let the step length u0 go to zero. The additional zeros off the y-axis from the stepwise approximation go then to complex infinity. This procedure of step-wise approximation of a function is very similar to the usual introduction of the definite integral.
If the sequence of numbers ( ) ( ) 0 1 n u ∆Ω + decreases rapidly in a way that the sum in (2.5) converges for all z ∈  then the function is an entire function of the complex variable i z x y = + . We will show that for arbitrary fixed real variable x it is a periodic function of the variable y. First, we find the periodicity ( ) ( ) with its (minimal) period length 0 2 u π and more generally From this according to (2.4) results the periodicity of the function with the same period length Although this is no more a periodicity of the functions ( ) In the next Section we derive some more detailed representations of the Xi which permit us to crystalize two different kinds of zeros in the considered step-wise commensurable approximation of the Omega function ( ) u Ω and call this in the further text shortly the "commensurable step-wise case".

Representation of the Xi Function in Commensurable Step-Wise Case by Chebyshev Polynomials
Since 0 z = is not a zero of the Xi function ( ) it is rational to investigate instead in most cases the function . We will show in next Section that in commensurable step-wise case there appear two different kinds of zeros from which the second kind is "not analytically" in a way which we will explain. Shortly saying, "not analytically" means that these zeros come from separate independent vanishing of two different functions which are not necessarily Real and Imaginary part of an analytic function of variable z. A great role plays in this case the periodicity condition (2.10).
The condition for zeros in the commensurable step-wise case is the simultaneous vanishing of Real and Imaginary part of (2.4) that means the vanishing of  We derive now alternative representations of this function which split from the Imaginary part a factor depending on ( ) , x y which is not involved in the vanishing of the zeros called "not analytically". This will be accomplished in next Section. For this purpose we use the Chebyshev polynomials of first and second kind and as preparation we consider some of their basic properties.
Using the following known identity of functions and are related to the Chebyshev ( ) Recurrence relations can be represented, for example, in the following forms (e.g., [21]) These relations hold for arbitrary complex variable t.

From (3.4) follows with substituted argument
x y x y x y x m x y n x y m n δ + π + π + + + The prolongation of these relations to negative 0 m < and (or) 0 n < is possible using the relations given in (3.4) and (3.5).
With (2.4) and (3.3) we have derived different representations of the Xi func- of commensurable step lengths and with sufficiently rapid decrease. We will now analyze their zeros in dependence on the step length 0 u . This can be completely done for the principal position of their zeros and is made in the next Section.

Two Kinds of Zeros in Commensurable Step-Wise Case
In ( The zeros which all lie on the imaginary axis are equidistant with half of the period length 0 2 k u π (see (2.10)). If 0 u goes to zero then this period length goes from its smallest zero at 1 together with all higher ones to complex in- Thus this kind of zeros vanish in the limiting case of vanishing step length 0 u from the list of zeros and is no more relevant for continuous Omega functions.
We consider now the second factor in (3.3) more in detail and write it as follows where we used the identity (3.5) for its transformation. Split in Real and Imaginary part we have Advances in Pure Mathematics it is only simple to establish a manageable condition for zeros on the imaginary axes 0 x = . In this case the second condition in (4.5) for vanishing of the Imaginary part is identically satisfied and we do not have to be concerned more about this condition. Therefore, for 0 x = there remains to be satisfied only the condition of vanishing of the Real part which simplifies in this case to ( ) on the imaginary axes are the only ones provides the Second mean-value or Bonnet approach which we give in Section 8. Now, however, we will examine the case of possible zeros in commensurable step-wise case off the imaginary axis.
We begin now to deal with the problem of zeros of ( ) z z Ξ off the imaginary axis in commensurable step-wise case. First, we derive a modified representation with separated Real and Imaginary part. If we exclude the zeros of the factor ( ) 0 sin u z which lie on the imaginary axis and are dealt with in (4.4) then for zeros to 0 x ≠ off the imaginary axis due to the inequality we have to satisfy the following two conditions for Real and Imaginary part of (4.7) The expressions on the two right-hand sides in (4.9) are not the Real and Imaginary part of an analytic function of variable i z x y = + . The zeros which we determine by the two conditions (4.9) are therefore "not analytically" determined.
We call them zeros of second kind for the commensurable step-wise kind and discuss them in next Section.

Possible Zeros of Xi Function in Commensurable
Step-Wise Case off the Imaginary Axis One case for a possible satisfaction of the second of the conditions (4.9) is immediately to see and is   following from (4.9). The right-hand sides of these two conditions are not Real and Imaginary part of an analytic function of the complex variable i z x y = + and thus it belongs to the case which we call "nonanalytic" ones. It is hardly possible to determine the zeros of the case (5.6) if such exist at all. However, it seems to us that we can prove that also these zeros for 0 0 u → go to complex infinity. For this purpose, we make in both conditions in (5.6) a Taylor series expansion of the Chebyshev polynomials in powers of 0 u according to Therefore in the limiting case 0 0 u → the conditions (5.6) cannot be satisfied if One may argument in favor of this result also in the following way. If for a certain finite 0 u exists a zero of both conditions (4.9) then in a higher approximation with 0 0 u u ′ < this zero will be destroyed and it may be expected that in the limiting case to vanishing 0 u it does not stabilize to a certain value.

Short Summary of the Second Mean-Value Approach to Zeros for Monotonically Decreasing Omega Functions
The approach to the zeros of the Riemann Xi function ( ) which for u → +∞ vanishes so rapidly that the integral exists was developed in [15]. We compile here the main results which seem to be correct and try to shut the gaps which were for the possible zeros off the imaginary axis.
The Second mean-value or Bonnet approach was applied in [15]  function is sufficient, see also Section 10) this should not be a cause that the Bonnet method cannot be applied. Since in case of convergence the integrand is an analytic function of the variable z the integral (1.6) should also be an analytic function of z and, therefore, also the mean-value function should be such. Thus the application of the Second mean-value or Bonnet approach to the integral (1.6) leads to the following principal form  Clearly, the same condition follows if we calculate the squared modulus of (6.5) and set it equal to zero which is only possible if both squares on the right-hand side vanish that leads again to the two conditions (6.6) for zeros. From the first condition (6.6) follows for zeros ( ) and as consequence separated for its Real and Imaginary part

Further Development of the Second Mean-Value Approach to Zeros of Monotonically Decreasing Omega Functions
We begin now to investigate the restrictions for zeros (6.6) in the Second meanvalue approach by derivation of some more technical relations which provide the extension of the functions in this approach from the imaginary axis i z y = to the whole complex z-plane.
In [15] was derived that an analytic function This is a consequence of the Cauchy-Riemann equations. Applied to the func- In this form they may be applied to the considered problem.
Due to vanishing of ( )

0
, v x y on the y-axis according to (6.11) for the func- we may directly apply relations (7.4) in the form However, we may apply them also to the function and for the Real part of ( ) 0 w z z 5 The analogous form was given for known functions ( ) One sign in the operator identities on the right-hand side of (7.6) and (7.7) has changed in comparison to (7.4). This happened since ( ) 0 0, u y y is now the Imaginary part of ( ) 0 w z z which alone is non-vanishing on the imaginary axis and from (7.3) follows for the reconstruction in this case The equivalence of the two forms (7.5) and (7.6) to (7.4) can be also established by the following general identities in application of the operators cos x y and sin x y to a product ( ) ( ) f y g y of two functions ( ) f y and ( ) and in similar way sin .
x f y g y x f y x g y y y y In their structure we find a striking similarity to the addition theorems for Cosine This may be considered as alternative proof of the relations (7.9) and (7.10) which play a role for the transition from the imaginary axis to axes parallel to the imaginary axis. In the following the eigenvalues and eigenfunctions of the operator i y ∂ − ∂ play a main role.

Principal Position of Analytic Zeros of Xi Functions in the Second Mean-Value Approach Including the Riemann Hypothesis
As two independent conditions for zeros of the function (6.5) in considered case of monotonically decreasing Omega functions including step-wise constant Omega functions with incommensurable step lengths we derived (6.6) using (7.6) and (7.7). The result may be written ( ) We now introduce the abbreviation Since usually we do not know this function explicitly we will make a general expansion of ( ) With this general assumption we now go into the conditions for zeros (8.1).
Due to orthonormality of the eigenfunctions in the Fourier expansion from these two equations follows with constants k a and k b and where we took into account the symmetry of Real and Imaginary part of the Xi function. These two solutions are contradictory and exclude themselves mutually. The only way out of this dilemma is to assume We excluded in this way all "analytical" zeros off the imaginary axis in limiting case of continuous Omega functions. Furthermore, for the limiting transition from the commensurable step-wise case all zeros off the imaginary axis go to complex infinity for step length 0 0 u → or do not provide any solution.
Since to the included cases of Omega functions belongs also the Omega function to the Riemann Xi function ( ) it seems to be a proof of the Riemann hypothesis that all zeros of the Riemann Xi function lie on the imaginary axis and thus also all nontrivial zeros of the Riemann zeta function ( ) s ζ . For the modified Bessel functions mentioned in Section 1 for which this property is already proved by their differential equations (e.g., [25] [26]) this would be an alternative proof. We think that the commensurable step-wise cases within all cases of monotonically decreasing Omega functions play a similar role as the rational numbers within all real numbers.

Some Further Remarks to the Derivations According to the Bonnet Approach
One may be astonished and it was an essential problem that in the Second mean-value approach we found according to (8.10) that all zeros lie on the imaginary axis whereas in the commensurable step-lengths approach we obtained also zeros off the imaginary axis. We could, however, show that for vanishing step lengths 0 0 u → the additional zeros off the imaginary axis go to complex infinity. For this purpose we followed the derivation of Second mean-value (Bonnet) approach in the cited monographs [16] [17] [18] and could not discover a defect in their correctness also for the case of the commensurable step-wise approach. They are derived in mentioned sources for real mean vales. We extended it to the complex analytic mean-value function ( ) 0 w z that is apparently justified for the whole complex integral (1.6) for continuous Omega functions but only not for the commensurable step-wise case where we could separate a factor ( ) 0 sh u z (see (3.3)) with its own zeros which are moreover periodically on the imaginary axis and the remaining functions with real and imaginary part cannot be unified to an analytic function. In this case as it seems to us one has to introduce two different mean-value function  We did not investigate this up to now. What was incorrect or even wrong in our article [15]! We obtained only the "analytic" zeros which are involved in ( ) ( ) 0 sh w z z z and, principally, lie on the imaginary axis i z y = . Moreover, this was not shown in such evident way as it is made in present paper in Section 8 by derivation of the contradiction (8.9) for the homogeneous solutions for the Fourier components of the function in the Hyperbolic Sine. The "non-analytic" solutions which are possible in the commensurable step-wise case (and, apparently, are possible only in this case as the limiting case 0 0 u → suggests) were not seen since they are not contained in the function (6.3). The operator identity 2 2 cos was applied there to the mean-value function was a faux pas and may be only used for identical transformations and simplifications but cannot provide new conclusions for the position of the zeros. However, using it one may see that the homogeneous equations to (8.6) or (8.7) are contradictory without providing a solution for 0 x ≠ . Our paper [19] was not intended as approach to a proof of the Riemann hypothesis and similar theorems for the modified Bessel function and was merely intended as illustration what one may expect for the zeros going from lower to higher approximations in Taylor series expansion of the Xi function but it was there shortly sketched the idea that one may establish for the commensurable step-wise case the exclusion of all additional zeros (now called "non-analytic" ones) off the imaginary axis and which go to complex infinity in the limiting case of vanishing step length. The execution of this plan in present paper proved to be much more difficult and branched than thought at this time.

Monotonic Decrease of Omega Function as Sufficient (or Necessary (or Both)) Condition for Zeros of Xi Functions Only on the Imaginary Axis
The monotonic decrease of the Omega function  where already ( ) ( ) 1 u Ω is not monotonically decreasing (or increasing) and is not even definite.
which are connected by the integral in (1.6). The Omega function is supposed to be a monotonically (non-strictly) decreasing function up to zero in infinity for u → ∞ and it is stated that the Xi function possesses zeros only on the imaginary axis y if the Omega function is not a step-wise constant function with commensurable step lengths. The step-wise constant Omega functions with commensurable step lengths are taken as approximations of the considered Xi function with final limiting procedure of step length 0 u going to zero where the zeros which do not lie on the imaginary axis and a part of additional zeros on the imaginary axis go to complex infinity. We show that due to splitting of their Xi function into the product (3.3) they possess two different kinds of zeros where the zeros of first factor are not restricted to the imaginary axis but go to complex infinity in the limiting case of vanishing step lengths whereas the zeros of the second factor stabilize to the "genuine" zeros of the considered function and are restricted to the imaginary axis. Since the made assumptions are true for the Omega function to the Riemann Xi function this proves if correct also the Riemann hypothesis that all non-trivial zeros of the Riemann zeta function lie on the axis parallel to the imaginary axis with real part 1 2 . This is accomplished in Section 8 by an, apparently simple, proof. Our approach is very similar to the primary introduction of the notion of a definite integral of a function by making first a step-wise approximation of the function and then going to zero with the step lengths by a limiting procedure. The further considerations of this article (Sections 6 -8) show that the treatment by the Second mean-value theorem is an alternative one to other approaches. To exclude the cases of Xi functions with zeros off the imaginary axis was absent in our article [15] and is here made. In Section 10, it is clarified which role plays the condition of monotonic decrease of the Omega function with the result that it is sufficient for presence of zeros only on the imaginary axis (under the other made assumptions) but is not necessary for this. We give examples with not monotonically decreasing Omega functions which lead to Xi functions with zeros only on the imaginary axis. In our examples these zeros are not simple zeros in all cases. In case of Omega functions which decrease in infinity only as simple exponential function (exponent proportional only to variable u) the integral (1.6) is no more convergent in the whole complex z-plane and there appear poles in the Xi function.
It seems to us that also an approach from the zeros of Taylor series approximations of the Xi function should be possible in case if it is perfectly possible to prove that in the transition to next higher approximations all zeros off the imaginary axis go to complex infinity and only the zeros on the imaginary axis stabilize as the "genuine" zeros of the Xi functions to continuous Omega functions with the supposed properties. The considered illustrations in [19] show such a behavior but a difficulty of the analytic treatment in this way is here that the transition from an approximation to the next higher approximation is hardly possible in perfect analytic form without further approximations leading finally

Appendix A
A category of proper and improper integrals over products of Hyperbolic and Trigonometric functions For the separate application of the second mean-value approach (Bonnet approach) to Real-and Imaginary part of (1.7) we derive here some proper and corresponding improper integrals where the last lead to Generalized functions.
The first two (proper) integrals with real parameter a which we need for the We calculate now the corresponding improper integrals for a → +∞ and start from the following two integrals which are well known from the theory of Fourier transformation of Generalized functions (e.g., [27], chap. II, §5, (15), (15')  where  applied to a function means that the principal value is to take at the singularity of the kind of a simple pole. As a consequence of these relations follows   We emphasize that such generalized functions as ( )