Common Properties of Riemann Zeta Function , Bessel Functions and Gauss Function Concerning Their Zeros

The behavior of the zeros in finite Taylor series approximations of the Riemann Xi function (to the zeta function), of modified Bessel functions and of the Gaussian (bell) function is investigated and illustrated in the complex domain by pictures. It can be seen how the zeros in finite approximations approach to the genuine zeros in the transition to higher-order approximation and in case of the Gaussian (bell) function that they go with great uniformity to infinity in the complex plane. A limiting transition from the modified Bessel functions to a Gaussian function is discussed and represented in pictures. In an Appendix a new building stone to a full proof of the Riemann hypothesis using the Second mean-value theorem is presented.


Introduction
The present paper tries to find out the common ground for the zeros of the .Concerning their zeros it is equivalent to the nontrivial zeros of the zeta function.In present paper we will mainly have to do only with this Xi function ( ) which we displaced in a way that its zeros lie on the imaginary axis provided; the Riemann Hypothesis is correct and we denote as Xi function The content of this article was not intended as a proof of the Riemann hypothesis but during the work we found a further, as it seems, essential building stone for its proof by the second mean-value approach which is represented in Appendix.The article is merely intended as an illustration to the zeros of a function with a possible representation in an integral form given in Section 2 (Equation (2.8)) with monotonically decreasing functions ( ) u Ω satisfied by the Riemann Xi function and by the modified Bessel functions and explains why the Gauss bell function although it can be represented in such form does not possess zeros.Other kinds of interesting illustrations of the Riemann zeta function (and of other functions) by the Newton flow are given in [9] [10].A main purpose was to understand how the zeros in the Taylor series approximations of such functions behave when we go from one order of the approximation to the next higher one.To get the possibility of a comparison with the pictures of zeros for functions without an integral representation of the mentioned form we made an analogous picture for an unorthodox entire function in Section 9.

Basic Equations for the Considered Functions
The Xi function ( ) z Ξ to the Riemann zeta function ( ) where ( ) is the Riemann Xi function [1] which is related to the Riemann zeta function ( ) The Riemann zeta function is basically defined by the following Euler product ( ) ( ) ( ) where n p is the sequence of prime numbers beginning with 1 2 p = .The definition of the Riemann Xi function (2.3) is equivalent to the definition by the following (Dirichlet) series for complex variable i z x y = + ( ) ( ) ( ) which is convergent for 1 x > and arbitrary y and can be analytically continued to the whole complex z-plane.( ) The functions ( ) are entire functions which satisfy the differential equation ( ) ( ) In comparison to ( ) exclude the zeros or infinities of the first ones at 0 z = but the other zeros remain the same for both functions.
Finally, we consider the Gaussian functions with parameter 2 0 a > which can be represented by the following integral representation (continued from the imaginary axis y to the whole complex z-plane) ( ) ( ) which become Gaussian bell functions for imaginary argument i z y = .Clearly, the functions do not possess zeros on the imaginary axis and zeros at all.
The three mentioned types of functions written as ( ) have in common that they are symmetrical functions in z and that they possess a representation by an integral of the type with monotonically decreasing functions ( ) The Taylor series of ( ) The odd moments and is independent of the chosen reference point.In the following the moments of the function play an important role.That

( )
u Ω is a symmetrical function in u is, in principle, not necessary since the integration over u in (2.8) is restricted by 0 u ≥ but the symmetry permits to extend the integral over negative values of u using it in the form if the integral exists in some sense (e.g.weak convergence).We consider this now more explicitly.
The explicit representation of the Xi function ( ) to the Riemann Xi function ( ) with the special values ( ) ( ) This was derived in detail in [11].The function together with its first derivative ( ) ( )  is not easily to see from (2.14) and it was a genuine surprise for us to meet such a kind of a symmetrical function (see discussion in [11]).The function ( ) , for example, is already not a symmetrical function for In case of the modified Bessel functions of imaginary argument the following basic integral representations are known which for the functions proportional to ( ) may be written as follows (taking into account or expressed by the Confluent Hypergeometric function ( ) The Taylor series expansion is The functions ( ) with the following explicit expressions for ( ) are equivalent to functions of 2 u only and are in this sense symmetrical functions of u and for real 2 1 u < we have to choose the real value of ( ) The first six cases of the function ( ) with integer or semi-integer index ( ) The function ( ) for the modified Bessel functions and also for the Gaussian bell function have in common that they are monotonically decreasing functions for 0 u ≥ up to ( ) 0 Ω +∞ = .

Modified Bessel Functions with Stretched Argument of the Kernel Function and Limiting Transition to Gaussian Bell Function
We now calculate the even moments ν Ω of the functions which lead to well-known integrals and in the special case 0 m = of zeroth moments ( ) These are the areas under the curves ( ) on the positive u-axes.The following considerations show that it is favorable to make ,0 ν Ω equal independently on parameter ν and to keep ( ) constant and we choose ( ) To keep in addition also ,0 ν Ω constant we have now the only possibility to introduce a stretch factor to the variable u and we make the transformation (now already with choice (3.3)) ( ) ( ) with the definition of from which follows for the zeroth moments of ( ) The functions ( ) Their Taylor series are The first three functions ( ) with integer and with semi-integer ν are explicitly The functions ( ) on the imaginary axes i z y = for integer and semi-integer index ν up to 9 2 ν = are illustrated in Figure 3.
is in visible way already hardly to distinguish from the Gaussian bell function (below we see to ).
That the mentioned approach for ν → ∞ to a Gaussian bell function is really true we establish exactly and determine these limits.
We now show that the new functions ( ) Gaussian function that becomes a Gaussian bell function for imaginary i z y = .
As auxiliary formulae for  (on the imaginary y-axis i z y = ) does not possess zeros.This Gaussian curve is not well to distinguish in the bulk of the other curves and is not drawn here.
and analogously for 1 Applying these approximations we find from (3.8) This means that in the limiting transition ν → ∞ the functions ( ) or by substitution i z y = with, in general, complex variable y where we use the identity →∞ Ω using the definition in (3.4) we find We may check the transition from ( ) via (2.8) using the auxiliary formula (2.7) with Furthermore, we find using the approximation (3.11) that the factor between 1 ,0 2 u and ,0 u ν in (3.5) approaches for 1 with a high precision and monotonically increasing without a finite limit.
We now show that for high increasing 1 ν  the first positive roots of ( ) Ξ and thus also the higher roots increase.For ( ) and it is well known that their roots ( ) Thus the first roots ( ) ν also to infinity.This explains more in detail why the first zeros ( ) in Figure 3 go to the limiting case We remind that the starting point for the derivation of the approximations was making equal the zeroth moments for ν → ∞ move in this process also to infinity but "very slowly".This is a good illustration why the Gaussian function which on the imaginary axis y becomes a Gaussian bell function does not possess zeros at all although its Omega function is monotonically decreasing.The first zero and in this way all other zeros are moved by the limiting transition to infinity although very slowly.The same is the case with the discontinuities of derivatives in the functions ( )

Graphical Representations to Zeros of the Xi Function to Riemann Zeta Function in Approximations by Truncated Taylor Series
In this Section we consider the Taylor series expansions of the Xi function The coefficients in this Taylor series are the even moments as defined.
We truncate the Taylor series of ( ) and calculate all zeros of these approximations up to a certain maximal M and make graphical representations of their zeros.As mentioned the coefficients of the series are determined by the even moments It is interesting to mention that the coefficients ( ) ( ) The first term in (4.3) can also be written (see [11]) In the truncation of the series in powers of z with the highest term proportional to 2M z we found by numerical solution of the corresponding algebraic equations of degree 2M the 2M complex solutions from which the 2 We made the calculations two times with "Mathematica 3" (up to 2M = 80 and 18 digits) and with "Mathematica 6" (up to 2M = 100 and 20 digits) with a time difference of some years.We did not get full agreement mainly in the last 5 digits in the higher coefficients and came already near to the limit of capabilities of our PC.
A In the following we give graphical illustrations of all zeros of the Taylor approximations in the complex z-plane where all zeros up to a certain order 2M are taken into account and where one may see how the zeros change from an order to a higher order.We explain first how the following figures are made.We take a certain order 2M of the Xi function ( ) given by the truncated Taylor series (4.2) and determine numerically all of its zeros and represents them by points in the complex ( ) -plane where we choose the same scale on the xand y-axis.Two variants are made, first the representation by isolated points and on the values given in the third and fourth column on the right-hand side did not become fully stable in our calculations in dependence on the number of sum terms taken into account in (2.14) and the chosen upper limit of integration that is rather due to the limits of our computer capabilities.The genuine value for the third root is near to 25.0109 instead of the stabilized 25.0101 seen in the table.The fourth root is at 30.4249.Therefore, the table reflects one such calculation.Onto the following graphical illustrations of zeros this does not have a visible influence.From the series (4.3) one may guess how difficult such calculations are even for a computer.second the representation by connected neighbored points.The obtained partial pictures are a little different for odd and even M that is represented in Figure 4 for 29 M = and 30 M = .
Then we calculate and represent all zeros of the Taylor approximations of ( ) to a certain maximal M and represent the zeros in described way by isolated points and by connected neighbored points in each of the approximations up to the maximal M.This is made in Figure 5 and  5 by some accumulation of points at these values.Figure 6 shows the same picture with all neighbored points in each approximation joined as described.Since not all details are well recognizable in Figure 6 the same is made in Figure 7 but  .The obtained 58, respectively, 60 zeros are shown in the complex plane as points without mutual distortion of the axis lengths.In the pictures to the right-hand sides we have joined neighbored numbers.On the imaginary axes where it is not clear which zeros are neighbored to zeros off the axis we went in clockwise sense on the positive part first to the highest zero and then to the next lower zeros and so on and then from the lowest zero on the imaginary axis clockwise to the next complex zero.This shows also the way we went in the next picture where the details on the imaginary axis are not so clearly visible.The two zeros at 1     only for the first 40 Taylor approximations where this is clearer to see.These pictures show that the zeros on the imaginary axis stabilize from order to higher orders at the genuine zeros of the Riemann zeta function on the y-axis and separate themselves from the main bulk of zeros in a considered order which do not lie on the imaginary axis.That this remains in this way for M → ∞ is, clearly, only a conjecture but in Section 8 we try to understand this by some analytic approximations.
In Figure 7 for the case 2 40 M = we see only one accumulation point of the zeros on the positive and negative y-axis corresponding to the two zeros One may check that generally For the moments of these functions result the inequalities ( The resulting function Clearly, as a Gaussian function it does not possess zeros on the y-axis and zeros at all.The ( ) u Ω function to the Riemann Xi function and the considered modified Bessel functions possess the common property that they vanish in infinity more rapidly (or are even finite) than the Omega function to a Gauss function.In principle, this does not exclude Omega functions which vanish less rapidly.For example, a function without zeros of ( ) at all in finite regions of the complex plane but with poles that regarding the zeros is the same as for a Gauss function (see Section 8 where this is explained by motion to infinity from finite approximations).However, by comparison with Gauss functions we may get inequalities for the moments of the considered Omega functions.

( ) z z sh
In this Section we now come back to the modified Bessel functions ( ) They are represented in Figure 8 and in Figure 9   .The neighbors within an approximations are not joined and it is not fully easy to see which point belongs to a certain approximation. x

y u x y x v x y y u x y y v x y x x y u x y x v x y y u x y y v x y x x y u x y x v x y y u x y y v x y x
From both forms of the right-hand side in (6.2) we find that for zeros of ( ) the following two conditions [11] (

, u x y y v x y x n n
have to be satisfied at the same time and this is necessary and sufficient.
We now consider the special case of ( ) z Ξ on the imaginary axis y and find from (6.2) Since due to symmetries (2.8) the function ( ) has to be a real-valued function that for 0 y ≠ in (6.4) is only possible if ( ) 0 0, v y vanishes we find on the imaginary axis This follows also immediately by application of the second mean-value theorem to the integral for ( ) where the mean value can only take on real values here in dependence on y as parameter.Thus for zeros on the imaginary axis ( ) the two conditions (6.3) reduce to only one condition ( ) The condition in the second line of (6.axis analytically or, at least, numerically but for many problems including the considered one it is not necessary to know these zeros explicitly. We return to the general case of general values x on the real axis with the conditions (6.3) for zeros.From (2.8) follows that ( ) should be an analytic function for all z for which the integral exists that means to satisfy the Cauchy-Riemann equations.For a general analytic function one can derive by integration of the Cauchy-Riemann equations the following relations in case of ( ) Now come into play the following operator identities (operator identities are such identities which can be applied to arbitrary functions to provide function identities) [11] cos cos sin , x y y x x x y y y More general identities of such kind can be derived by representing the Cosine and Sine functions by Exponential functions and using that exp ix y applied to analytic functions ( ) f y displace the argument of these functions to ( ) ± that is discussed in [11].Using (6.7) and (6.8) we may write the conditions for zeros (6.3) in the following way ( ) ( ) If we now apply the operator cos x y to the first of these conditions and the operator sin x y to the second one we obtain ( ) ( ) By addition of both equalities using the operator identity follows from (6.10) This consequence for zeros results from both conditions (6.3) and provides a and have to look for solutions x at the same time of both equations ( ) ( ) If one finds such solutions is stepwise constant and monotonically decreasing with equal lengths 0 u of the steps and this is the only case (Appendix A).
The corresponding functions ( )  with different amplitudes.This was not correctly discussed in [11] and Katsnelson [16] showed an error in a short Email 4 and we recognized it [17] but it was already seen in [11] that the possible zeros off the imaginary axis, i.e. 0 0 0 i z x y = ± of ( ) with 0 0 x ≠ , must possess imaginary parts 0 y which agree with one of the zeros on the imaginary axis.Reactions regarding concern about the applicability of the Bonnet theorem for present case were expressed by others, in particular, in a nice Email by Gélinas [18] with appended file but I could not find it published now.This also means that the Riemann hypothesis which is the absence of zeros of the Riemann Xi function off the imaginary axis through 1 2 x = was not correctly solved by the second mean-value approach to this time although it was very improbable that a nontrivial zero of the Riemann zeta function off the imaginary axis has exactly the same imaginary value as that of one on the imaginary axis through 1 2 x = .
We add now in Appendix A a further important building stone to a full proof of the Riemann hypothesis by the second mean-value theorem which seems to be 4 The full text in the Email from 06.01.2017 with the subject line "Riemannsche Vermutung" was the following: "The result is wrong.Counterexample:

Zeros of Taylor Series Approximations of Gauss Function and Absence of Genuine Zeros Understood in Uncommon Way
We now consider the Gauss function and calculate all zeros of its low-order Taylor series approximations ( )

Approximations of the Zeros from One to the Next Higher Orders
The zeros in each approximation for the considered functions are either pairs ( ) ( ) on the imaginary axis or in majority quadruples and there was no doubt which are pairs and which are quadruples even in case that their real part k x is small compared with maximal modulus of the zeros and since the whole number of zeros has to be 2M.It is noticeable that new zeros on the imaginary axis when they first appear in the 2M-th approximation may disappear in the next higher ( ) 2 2 M + -th approximation from the imaginary axis and reappear then in the ( ) 2 4 M + -th approximation as can be also seen from Table 1.In every case when there appeared a new zero on the imaginary axis the next lower zeros began to stabilize and to decouple from the main bulk of zeros in the complex domain and stabilize there in each new ( ) exp z the zeros on the imaginary axis show a similar picture with alternatingly generating and not generating zeros on the imaginary axis from one approximation 2M to the next higher approximation 2 2 M + .The main bulk of zeros in the complex domain in all these pictures drifts with their modulus to infinity although very slowly that we can see in Figure 11.In the other pictures this is the main bulk of zeros which does not correspond to genuine zeros of the considered functions whereas the lower zeros on the imaginary axis stabilized more and more to the genuine zeros.
To understand the discussed behavior of the zeros from order to next higher order we try to discuss this now in some approximation.We suppose that we have the Taylor series approximation of a Xi function ( ) We assume that 0 z is an exact zero in 2M-th approximation that means a solution of the equation Then we try to calculate next higher solutions 0 z z z = + ∆ which are near to 0 z from the next higher 2 2 M + approximation which satisfies the equation In full generality this equation would provide the 2 2 M + solutions of zeros from only one arbitrary solution 0 z in the considered approximation but in such generality we cannot and do not want to solve it.Well soluble is the equation for additions to 0 z leading to a quadratic equation for z ∆ .This provides two solutions in a neighborhood for the considered 0 z .Thus we now make an expansion of the left-hand side in (8.3) If we neglect from the additional terms of the ( ) as small terms we obtain the following quadratic equation for The two solutions of this equation are or more compactly written where ( ) ( ) One may express this also by the logarithmic derivative of ( ) ( ) that, however, is inconvenient since it goes into the formula as a denominator.
For the most interesting case of points 0 0 i z y = on the imaginary axis we find from (8.6)The sum terms in braces are real ones but likely change their signs at the zeros (we think that it can be proved).The first sum term in braces changes a little the imaginary value of the root but the second term with the root with the two possible signs must be real or imaginary.In case of negative values of the content of the root it gives two imaginary values and together with the whole expression it provides two corrections off the imaginary axis.This can be seen in many of the picture for the roots.However a full discussion of the behavior from approximation to next higher approximation by (8.8) is complicated and has to describe how the roots stabilize on the imaginary axis in dependence on the moments.Such a discussion we cannot give to this time.

About the Zeros of an Unorthodox Function in Their Taylor Series Approximations
We consider here shortly for comparison with the pictures for the up to now discussed functions with a representation of the principal form (2. with respect to the zeros in its finite Taylor series approximations.This function plays a role for the calculation of the properties of coherent phase states [19]. The function ( ) f z in (9.1) possesses even and odd powers of variable z and, therefore it is not symmetrical with mirror symmetry to the real and imaginary axis but only symmetrical to the real axis.Taking separately the even and odd powers of z and applying the duplication formula for the factorials one may represent (9.1) in the form ( ) ( ) with the possible approximations in the coefficients for 1 m  (see (3.11)) This shows that ( ) In Figure 12 we see the first three pairs of zeros as some accumulation points.
The stable zeros on the imaginary axis under variation of u 0 are determined by ( ) 0 π, 1, 2, u y m m = = ± ±  but due to symmetry y y ↔ − we discuss sometimes so as if we take into account only positive y.We now bring the factor z) for imaginary argument z and, furthermore, for the absence of zeros of the Gaussian Bell function ( ) 2 exp z .For the function now called Riemann zeta function ( ) z ζ which was known already to Euler but was extended by Riemann to the complex plane Riemann expressed the hypothesis that all nontrivial zeros of this function lie on the axis 1 the imaginary axis y (Riemann hypothesis) [1] [2] [3] (both with republication of Riemann's paper) and many others, e.g.[4] [5] [6] [7] [8].Riemann never proved his hypothesis.He introduced in [1] also a Xi function ( ) z ξ which excludes the only singularity of the function ( ) z = − −  and possesses more symmetry than the zeta function ( ) z ζ

.
With respect to the position of the zeros the function ( ) z Ξ is fully equivalent to the nontrivial zeros of the Riemann zeta function ( ) z ζ only with displacement of the imaginary axis to these zeros.

1 z
= and its "trivial" zeros at 2, 4, 6, z = − − −  .Next we consider the whole class of modified Bessel functions is connected with the basic class of Bessel functions ( ) J z ν in the following slightly modified form by ( rapidly for u → ±∞ .The function ( ) z Ξ increases rapidly for real z x = → ±∞ and decreases rapidly for imaginary i i z y = → ± ∞ .The symmetry A. Wünsche DOI: 10.4236/apm.2019.93013285 Advances in Pure Mathematics

Figure 1
Figure 1.Representation of ( ) u Ω and of its first derivative

Figure 2 and
Figure 2 and Figure 3 admit the conjecture that

2
the Bessel functions is stretched by the factor ,0 u ν and therefore on the imaginary axis we have the functions diminishes the values for the roots (3.18) by the factor the Riemann Xi function ( ) s ξ in powers of z defined in (2.14) according to , are ("slowly") monotonically increasing for 21 42 m > =.
splitting of terms in the representation (2.14) of ( )

Figure 6
Figure 6 for maximal 2 60 M = .These approximations capture already approximations of the first two nontrivial zeros of the Riemann zeta function on the positive y-axis at 1 14.135 y ≈ and at 2 21.022 y ≈ seen in Figure 5 by

Figure 5 .
Figure 5. Zeros of Xi function ( ) z Ξ to Riemann zeta function in the first 30 approximation

Figure 6 .
Figure 6.Zeros of Xi function

∑
of its Taylor series with 2 2,4, ,60 M =  .The neighbored zeros are joined in each approximation separately.In the immediate neighborhood of the axis the picture becomes a little confusing since then happens a big step to the smallest genuine zero of the Xi function and then rises up to higher zeros on the imaginary axis before it goes to the main bulk of zeros outside the imaginary axis.

Figure 7 .
Figure 7. Zeros of Xi function( ) z Ξ to Riemann zeta function in the first 20 approximation . The four higher zeros on the positive y-axis (correspondingly negative y-axis) belong already to not yet stabilized approximations to the second amplitude and the stretching of the parameter u are chosen in the way that the first two terms of the Taylor series approximation are equal.For the function( )

Figure 8 .
Figure 8. Zeros of the function ( ) ( ) ( )2 3) is then identically satisfied.If one knows the function ( ) 0 0, u y one may determine the zeros on the imaginary A. Wünsche DOI: 10.4236/apm.2019.93013302 Advances in Pure Mathematics necessary condition for the imaginary values y of all zeros for which their imaginary value has to agree with one of the solutions for zeros on the imaginary axis.Thus we have to take all solutions y for zeros on the imaginary axis which we denote now by 0 y and which satisfy the conditions

2 2 M
Figure 8 and Figure 9 for a modified Bessel function 8) the following unorthodox entire function of an essential other kind

12 .
and, clearly, possesses the same zeros as the function ( )f z .We now give a graphical representation of the zeros of the Taylor series approximations Apparently, the computer calculated correctly up to this high approximation that we judged only from the optical impression of the figure in comparison to figures of such kind for smaller values N. To join neighbored points of each Taylor series approximation, at least, for such high maximal N becomes unfavorable.

Figure 12 .., 3 3
Figure 12.Zeros of unorthodox though entire function ( ) ) and for the limiting transition to a Gaussian function and compare this with the zeros for the Xi function(2.14)to the Riemann zeta function.In each
one of the conditions then it is almost in all cases not a solution of the other condition and, therefore, 0