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

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Common Properties of Riemann Zeta Function, Bessel Functions and Gauss Function Concerning Their Zeros Advances in Pure Mathematics, 2019, 9, 281-316 http://www.scirp.org/journal/apm ISSN Online: 2160-0384 ISSN Print: 2160-0368 Common Properties of Riemann Zeta Function, Bessel Functions and Gauss Function Concerning Their Zeros Alfred Wünsche Institut für Physik, Humboldt-Universität, MPG Nichtklassische Strahlung, Berlin, Germany How to cite this paper: Wünsche, A. Abstract (2019) Common Properties of Riemann Zeta Function, Bessel Functions and Gauss The behavior of the zeros in finite Taylor series approximations of the Rie- Function Concerning Their Zeros. Ad- mann Xi function (to the zeta function), of modified Bessel functions and of vances in Pure Mathematics, 9, 281-316. the Gaussian (bell) function is investigated and illustrated in the complex https://doi.org/10.4236/apm.2019.93013 domain by pictures. It can be seen how the zeros in finite approximations ap- Received: March 1, 2019 proach to the genuine zeros in the transition to higher-order approximation Accepted: March 26, 2019 and in case of the Gaussian (bell) function that they go with great uniformity Published: March 29, 2019 to infinity in the complex plane. A limiting transition from the modified Bes- Copyright © 2019 by author(s) and sel functions to a Gaussian function is discussed and represented in pictures. Scientific Research Publishing Inc. In an Appendix a new building stone to a full proof of the Riemann hypothe- This work is licensed under the Creative sis using the Second mean-value theorem is presented. Commons Attribution International License (CC BY 4.0). Keywords http://creativecommons.org/licenses/by/4.0/ Open Access Riemann Zeta and Xi Function, Modified Bessel Functions, Second Mean-Value Theorem or Gauss-Bonnet Theorem, Riemann Hypothesis 1. Introduction The present paper tries to find out the common ground for the zeros of the Riemann zeta function ζζ( z) =( xy + i ) and of the modified Bessel functions Iν ( z) (or Bessel functions Jν ( z) of imaginary argument z) for imaginary argument z and, furthermore, for the absence of zeros of the Gaussian Bell function exp( z2 ) . For the function now called Riemann zeta function ζ ( z) 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 1 1 lie on the axis zy= + i that means on the axis through x = and parallel to 2 2 DOI: 10.4236/apm.2019.93013 Mar. 29, 2019 281 Advances in Pure Mathematics A. Wünsche 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 ξ ( z) which excludes the only singularity of the function ζ ( z) at z = 1 and its trivial zeros at z =−−2, 4, and possesses more symmetry than the zeta function ζ ( z) . 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 ξ ( z) 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 Ξ( z) . With respect to the position of the zeros the function Ξ( z) is fully equivalent to the nontrivial zeros of the Riemann zeta function ζ ( z) only with displacement of the imaginary axis to these zeros. 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 Ω(u) 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. 2. Basic Equations for the Considered Functions The Xi function Ξ( z) to the Riemann zeta function ζ ( z) is defined by 1 Ξ≡( zz) ξ +, (2.1) 2 where ζ ( z) is the Riemann Xi function [1] which is related to the Riemann 1 zeta function ζ ( z) by z z − ξζ( zz) ≡−( 1!)π 2 ( z). (2.2) 2 The Riemann zeta function is basically defined by the following Euler product −1 ∞ 1 ζ ≡− ==== ( z) ∏ 1z ,( pppp1234 2, 3, 5, 7,) , (2.3) = n 1 ( pn ) 1Riemann denotes the complex variable by st=σ + i that is ξ (s) for the Xi function and ζ (s) for the zeta function. DOI: 10.4236/apm.2019.93013 282 Advances in Pure Mathematics A. Wünsche where pn is the sequence of prime numbers beginning with p1 = 2 . The definition of the Riemann Xi function (2.3) is equivalent to the definition by the following (Dirichlet) series for complex variable zxy= + i ∞ 1 ζ = = > ( z) ∑ z ,( xz Re( ) 1) , (2.4) n=1 n which is convergent for x > 1 and arbitrary y and can be analytically continued to the whole complex z-plane. The function Ξ( z) is an entire function which excludes the only singularity of ζ ( z) at z = 1 and its “trivial” zeros at z =−−−2, 4, 6, . Next we consider the whole class of modified Bessel functions Iν ( z) of imaginary argument which is connected with the basic class of Bessel functions Jν ( z) in the following slightly modified form by ( µµ!1≡Γ( + ) ) νν 2m 22 ∞ 1z Iνν( zz) = Ji( ) ≡∈∑ ,.(ν ) (2.5) z i z m=0 mm!( +ν ) !2 ν 2 The functions Iν ( z) are entire functions which satisfy the differential z equation 22 ν ∂∂2 2 ∂22 z+2νν zz − I0νν( z) = ⇔ z −− zI0.( z) = (2.6) ∂∂z zz ∂z ν 2 In comparison to Iν ( z) the functions Iν ( z) exclude the zeros or z infinities of the first ones at z = 0 but the other zeros remain the same for both functions. az22 Finally, we consider the Gaussian functions exp with parameter 4 a2 > 0 which can be represented by the following integral representation (continued from the imaginary axis y to the whole complex z-plane) 22 2 az 2 +∞ u =−>2 exp ∫ du exp2 ch(uz) ,( a 0) , (2.7) 4 πa2 0 a which become Gaussian bell functions for imaginary argument zy= i . Clearly, az22 the functions exp do not possess zeros on the imaginary axis and zeros 4 at all. The three mentioned types of functions written as Ξ( z) have in common that they are symmetrical functions in z and that they possess a representation by an integral of the type +∞ * Ξ( z) =d u Ω( u) ch( uz) ) =Ξ−( z) =Ξ z* , Ω( u) =Ω−( u), (2.8) ∫0 ( ( )) with monotonically decreasing functions Ω(u) for 0 ≤u ≤∞ that means 0 ≤uu12 ≤ ⇒ Ω( u1) ≥ Ω( u 2), Ω( u → +∞) = 0. (2.9) DOI: 10.4236/apm.2019.93013 283 Advances in Pure Mathematics A. Wünsche The Taylor series of Ξ( z) can be written in the form ∞ 11+∞ Ξz = Ω z2mn,d Ω ≡u Ω u +Ω − uu ( ) ∑ 2mn∫−∞ ( ( ) ( )) m=0 n!2 +∞ 1 2m Ω=du Ω( uu) , Ω+ = 0,(m = 0,1,2,) , (2.10) 2mm(2!m) ∫0 21 where Ω=n ,(n 0,1,2,) are defined as the moments of the symmetrical function Ω(u) with respect to the reference point u0 = 0 . A consequence of the definitions in (2.8) and (2.10) is +∞ Ξ(0d) =uu Ω( ) ≡Ω . (2.11) ∫0 0 The odd moments Ω21m+ of the function Ω(u) in the definition (2.10) vanish. The function Ξ( z) at z = 0 is equal to the zeroth moment Ω0 of the function Ω(u) and is independent of the chosen reference point. In the following the moments of the function Ω(u) play an important role. That Ω(u) is a symmetrical function in u is, in principle, not necessary since the integration over u in (2.8) is restricted by u ≥ 0 but the symmetry permits to extend the integral over negative values of u using it in the form 11+∞ +∞ 1+∞ Ξ=zde u Ω u uz = dchdsh,u Ω u uz + u Ω u uz ( ) ∫∫−∞ ( ) −∞ ( ) ( ) ∫−∞ ( ) ( ) (2.12) 22 2 =0 which for imaginary zy= i is a Fourier transformation of Ω(u) with the inversion 1 +∞ − Ω=(u) d yy Ξ( ie) iuy , (2.13) π ∫−∞ if the integral exists in some sense (e.g. weak convergence). We consider this now more explicitly. The explicit representation of the Xi function Ξ( z) to the Riemann Xi 1 function ξ ( zz) =Ξ− in this form (2.8) together with (2.9) is 2 +∞ Ξ=( z) d u Ω( u) ch( uz) , ∫0 ∞ u 22uu 22 22u Ω(u) ≡4exp∑ π nne2( π e− 3) exp( −π ne,) =Ω−( u) (2.14) 2 n=1 with the special values +∞ 11 Ξ(0) = duu Ω( ) ≈ 0.4971207782, Ξ± = , ∫0 22 Ω≈(0) 1.7867876019.
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