Approach to a Proof of the Riemann Hypothesis by the Second Mean-Value Theorem of Calculus

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1 Advances in Pre Mathematics, 6, 6, ISSN Online: ISSN Print: Approach to a Proof of the Riemann Hypothesis by the Second Mean-Vale Theorem of Calcls Alfred Wünsche Hmboldt-Universität Berlin, Institt für Physik, Nichtklassische Strahlng (MPG), Berlin, Germany How to cite this paper: Wünsche, A. (6) Approach to a Proof of the Riemann Hypothesis by the Second Mean-Vale Theorem of Calcls. Advances in Pre Mathematics, 6, Received: Jly 9, 6 Accepted: December 7, 6 Pblished: December, 6 Copyright 6 by athor and Scientific Research Pblishing Inc. This work is licensed nder the Creative Commons Attribtion International License (CC BY 4.). Open Access Abstract By the second mean-vale theorem of calcls (Gass-Bonnet theorem) we prove that the class of fnctions Ξ ( z) with an integral representation of the form d Ω( ) ch ( z ) with a real-valed fnction Ω( ) which is non-increasing and decreases in infinity more rapidly than any exponential fnctions exp ( λ), λ > possesses zeros only on the imaginary axis. The Riemann zeta fnction ζ ( s) as it is known can be related to an entire fnction ξ ( s) with the same non-trivial zeros as ζ ( s). Then after a trivial argment displacement s z = s we relate it Ξ z = d Ω ch z to a fnction Ξ ( z) with a representation of the form where Ω ( ) is rapidly decreasing in infinity and satisfies all reqirements necessary for the given proof of the position of its zeros on the imaginary axis z = iy by the second mean-vale theorem. Besides this theorem we apply the Cachy- Riemann differential eqation in an integrated operator form derived in the Appendix B. All this means that we prove a theorem for zeros of Ξ ( z) on the imaginary axis z = iy for a whole class of fnction Ω ( ) which incldes in this way the proof of the Riemann hypothesis. This whole class incldes, in particlar, also the modified Bessel fnctions Iν ( z) for which it is known that their zeros lie on the imaginary axis and which affirms or conclsions that we intend to pblish at another place. In the same way a class of almost-periodic fnctions to piece-wise constant non-increasing fnctions Ω ( ) belong also to this case. At the end we give shortly an eqivalent way of a more formal description of the obtained reslts sing the Mellin transform of fnctions with its variable sbstitted by an operator. Keywords Riemann Hypothesis, Riemann Zeta Fnction, Xi Fnction, Gass-Bonnet DOI:.436/apm December, 6

2 Theorem, Mellin Transformation. Introdction The Riemann zeta fnction ζ ( s) which basically was known already to Eler establishes the most important link between nmber theory and analysis. The proof of the Riemann hypothesis is a longstanding problem since it was formlated by Riemann [] in 859. The Riemann hypothesis is the conjectre that all nontrivial zeros of the Riemann zeta fnction ζ ( s) for complex s = σ + it are positioned on the line s = + i t that means on the line parallel to the imaginary axis throgh real vale σ = in the complex plane and in extension that all zeros are simple zeros []-[7] (with extensive lists of references in some of the cited sorces, e.g., ([4] [5] [9] [] [4]). The book of Edwards [5] is one of the best older sorces concerning most problems connected with the Riemann zeta fnction. There are also mathematical tables and chapters in works abot Special fnctions which contain information abot the Riemann zeta fnction and abot nmber analysis, e.g., Whittaker and Watson [] (chap. 3), Bateman and Erdélyi [8] (chap. ) abot zeta fnctions and [9] (chap. 7) abot nmber analysis, and Apostol [] [] (chaps. 5 and 7). The book of Borwein, Choi, Rooney and Weirathmeller [] gives on the first 9 pages a short accont abot achievements concerning the Riemann hypothesis and its conseqences for nmber theory and on the following abot 4 pages it reprints important original papers and expert witnesses in the field. Riemann has pt aside the search for a proof of his hypothesis after some fleeting vain attempts and emphasizes that it is not necessary for the immediate objections of his investigations [] (see [5]). The Riemann hypothesis was taken by Hilbert as the 8-th problem in his representation of 3 fndamental nsolved problems in pre mathematics and axiomatic physics in a lectre hold on 8 Agst in 9 at the Second Congress of Mathematicians in Paris [] [3]. The vast experience with the Riemann zeta fnction in the past and the progress in nmerical calclations of the zeros (see, e.g., [5] [] [] [6] [7] [4] [5]) which all confirmed the Riemann hypothesis sggest that it shold be tre corresponding to the opinion of most of the specialists in this field bt not of all specialists (argments for dobt are discssed in [6]). The Riemann hypothesis is very important for prime nmber theory and a nmber of conseqences is derived nder the nproven assmption that it is tre. As already said a main role plays a fnction ζ ( s) which was known already to Eler for real variables s in its prodct representation (Eler prodct) and in its series representation (now a Dirichlet series) and was contined to the whole complex s -plane by Riemann and is now called Riemann zeta fnction. The Riemann hypothesis as said is the conjectre that all nontrivial zeros of the zeta fnction ζ ( s) lie on the axis 973

3 parallel to the imaginary axis and intersecting the real axis at s =. For the tre hypothesis the representation of the Riemann zeta fnction after exclsion of its only singlarity at s = and of the trivial zeros at s = n, ( n =,, ) on the negative real axis is possible by a Weierstrass prodct with factors which only vanish on the critical line σ =. The fnction which is best sited for this prpose is the so-called xi fnction ξ ( s) which is closely related to the zeta fnction ζ ( s) and which was also introdced by Riemann []. It contains all information abot the nontrivial zeros and only the exact positions of the zeros on this line are not yet given then by a closed formla which, likely, is hardly to find explicitly bt an approximation for its density was conjectred already by Riemann [] and proved by von Mangoldt [7]. The (psedo)-random character of this distribtion of zeros on the critical line remembers somehow the (psedo)-random character of the distribtion of primes where one of the differences is that the distribtion of primes within the natral nmbers becomes less dense with increasing integers whereas the distribtions of zeros of the zeta fnction on the critical line becomes more dense with higher absolte vales with slow increase and approaches to a logarithmic fnction in infinity. There are new ideas for analogies to and application of the Riemann zeta fnction in other regions of mathematics and physics. One direction is the theory of random matrices [6] [4] which shows analogies in their eigenvales to the distribtion of the nontrivial zeros of the Riemann zeta fnction. Another interesting idea fonded by Voronin [8] (see also [6] [9]) is the niversality of this fnction in the sense that each holomorphic fnction withot zeros and poles in a certain circle with radis less can be approximated with arbitrary reqired accrateness in a small domain of the zeta fnction to the right of the critical line within s. An interesting idea is elaborated in articles of Neberger, Feiler, Maier and Schleich [3] [3]. They consider a simple first-order ordinary differential eqation with a real variable t (say the time) for given arbitrary analytic fnctions f ( z ) where the time evoltion of the fnction for every point z finally transforms the fnction in one of the zeros f ( z ) = of this fnction in the complex z -plane and illstrate this process graphically by flow crves which they call Newton flow and which show in addition to the zeros the separatrices of the regions of attraction to the zeros. Among many other fnctions they apply this to the Riemann zeta fnction ζ ( z) in different domains of the complex plane. Whether, however, this may lead also to a proof of the Riemann hypothesis is more than qestionable. Nmber analysis defines some fnctions of a continos variable, for example, the nmber of primes π ( x) less a given real nmber x which last is connected with the discrete prime nmber distribtion (e.g., [3] [4] [5] [7] [9] []) and establishes the connection to the Riemann zeta fnction ζ ( s). Apart from the prodct representation of the Riemann zeta fnction the representation by a type of series which is 974

4 now called Dirichlet series was already known to Eler. With these Dirichlet series in nmber theory are connected some discrete fnctions over the positive integers n =,, which play a role as coefficients in these series and are called arithmetic fnctions (see, e.g., Chandrasekharan [4] and Apostol [3]). Sch fnctions are the Möbis fnction µ ( n) and the Mangoldt fnction Λ ( n) as the best known ones. A short representation of the connection of the Riemann zeta fnction to nmber analysis and of some of the fnctions defined there became now standard in many monographs abot complex analysis (e.g., [5]). Or means for the proof of the Riemann hypothesis in present article are more conventional and old-fashioned ones, i.e. the Real Analysis and the Theory of Complex Fnctions which were developed already for a long time. The most promising way for a proof of the Riemann hypothesis as it seemed to s in past is via the already mentioned entire fnction ξ ( s) which is closely related to the Riemann zeta fnction ζ ( s). It contains all important elements and information of the last bt excldes its trivial zeros and its only singlarity and, moreover, possesses remarkable symmetries which facilitate the work with it compared with the Riemann zeta fnction. This fnction ξ ( s) was already introdced by Riemann [] and dealt with, for example, in the classical books of Titchmarsh [3], Edwards [5] and in almost all of the sorces cited ξ s and at the beginning. Present article is mainly concerned with this xi fnction its investigation in which, for convenience, we displace the imaginary axis by to the right that means to the critical line and call this Xi fnction Ξ ( z) with z = x+ iy. We derive some representations for it among them novel ones and discss its properties, inclding its derivatives, its specialization to the critical line and some other featres. We make an approach to this fnction via the second mean vale theorem of analysis (Gass-Bonnet theorem, e.g., [37] [38]) and then we apply an operator identity for analytic fnctions which is derived in Appendix B and which is eqivalent to a somehow integrated form of the Cachy-Riemann eqations. This among other not so sccessfl trials (e.g., via moments of fnction Ω ( ) ) led s finally to a proof of the Riemann hypothesis embedded into a proof for a more general class of fnctions. Or approach to a proof of the Riemann hypothesis in this article in rogh steps is as follows: First we shortly represent the transition from the Riemann zeta fnction ζ ( s) of complex variable s = σ + it to the xi fnction ξ ( s) introdced already by Riemann and derive for it by means of the Poisson smmation formla a representation which is convergent in the whole complex plane (Section with main formal part in Appendix A). Then we displace the imaginary axis of variable s to the critical line at s = + i t by s z = s that is prely for convenience of frther working with the formlae. However, this has also the desired sbsidiary effect that it brings s into the fairway of the complex analysis sally represented with the complex variable z = x+ iy. The s Ξ z fnction. transformed ξ fnction is called 975

5 Ξ is represented as an integral transform of a real-valed fnction d ch which is related to a Forier transform (more exactly to Cosine Forier transform). If the Riemann Ξ z occr for The fnction ( z) Ω ( ) of the real variable in the form Ξ ( z ) = Ω( ) ( z ) hypothesis is tre then we have to prove that all zeros of the fnction x =. To the Xi fnction in mentioned integral transform we apply the second mean-vale theorem of real analysis first on the imaginary axes and discss then its extension from the imaginary axis to the whole complex plane. For this prpose we derive in Appendix B in operator form general relations which allow to extend a holomorphic fnction from the vales on the imaginary axis (or also real axis) to the whole complex plane which are eqivalents in integral form to the Cachy-Riemann eqations in differential form and apply this in specific form to the Xi fnction and, more precisely, to the mean-vale fnction on the imaginary axis (Sections 3 and 4). Then in Section 5 we accomplish the proof with the discssion and soltion of the two most important eqations () and () for the last as decisive stage of the proof. These two eqations are derived in preparation before this last stage of the proof. From these eqations it is seen that the obtained two real eqations admit zeros of the Xi fnction only on the imaginary axis. This proves the Riemann hypothesis by the eqivalence of the Riemann zeta fnction ζ ( s) to the Xi fnction Ξ ( z) and embeds it into a whole class of fnctions with similar properties and positions of their zeros. The Sections 6-7 serve for illstrations and graphical representations of the specific parameters (e.g., mean-vale parameters) for the Xi fnction to the Riemann hypothesis and for other fnctions which in or proof by the second mean-vale problem are inclded for the existence of zeros only on the imaginary axis. This is, in particlar, also the whole class of modified Bessel fnctions Iν ( z), < ν < with real indices ν which possess zeros only on the imaginary axis y and where a proof by means of the differential eqations exists and certain classes of almost-periodic fnctions. We intend to present this last topics in detail in ftre.. From Riemann Zeta Fnction ζ ( s ) to Related Xi Fnction ζ ( s ) and Its Argment Displacement to Fnction Ξ ( z) In this Section we represent the known transition from the Riemann zeta fnction ζ ( s) to a fnction ξ ( s) and finally to a fnction Ξ ( z) with displaced complex variable s z = s for rational effective work and establish some of the basic representations of these fnctions, in particlar, a kind of modified Cosine Forier transformations of a fnction Ω ( ) to the fnction Ξ ( z). As already expressed in the Introdction, the most promising way for a proof of the Riemann hypothesis as it seems to s is the way via a certain integral representation of the related xi fnction ξ ( s). We sketch here the transition from the Riemann zeta s ξ s in a short way becase, in principle, fnction ζ to the related xi fnction 976

6 it is known and we delegate some aspects of the derivations to Appendix A. Usally, the starting point for the introdction of the Riemann zeta fnction ζ ( s) is the following relation between the Eler prodct and an infinite series contined to the whole complex s -plane ζ, Re, (.) s s k = pk n= n ( s) = ( σ ( s) > ) where p k denotes the ordered seqence of primes ( p =, p = 3, p3 = 5, ). The transition from the prodct formla to the sm representation in (.) via transition to the Logarithm of ζ ( s) and Taylor series expansion of the factors log s in p k s powers of p k sing the niqeness of the prime-nmber decomposition is well known and de to Eler in 737. It leads to a special case of a kind of series later introdced and investigated in more general form and called Dirichlet series. The Riemann zeta fnction ζ ( s) can be analytically contined into the whole complex plane to a meromorphic fnction that was made and sed by Riemann. The sm in (.) converges niformly for complex variable s = σ + it in the open semi-planes with arbitrary σ > and arbitrary t. The only singlarity of the fnction ζ ( s) is a simple pole at s = with reside that we discss below. The prodct form (.) of the zeta fnction ζ ( s) shows that it involves all prime nmbers p n exactly one times and therefore it contains information abot them in a coded form. It proves to be possible to regain information abot the prime nmber distribtion from this fnction. For many prposes it is easier to work with meromorphic and, moreover, entire fnctions than with infinite seqences of nmbers bt in first case one has to know the properties of these fnctions which are determined by their zeros and their singlarities together with their mltiplicity. From the well-known integral representation of the Gamma fnction z t ( z) tt ( ( z) ) Γ = d e, Re >, (.) µ follows by the sbstittions t = n x, µ z = s with an appropriately fixed parameter µ > for arbitrary natral nmbers n s µ µ n x s xx s µ = d e, Re. s n > (.3) Γ µ Inserting this into the sm representation (.) and changing the order of smmation and integration, we obtain for choice µ = of the parameter sing the sm evalation of the geometric series s p ζ ( s) = d p, ( Re( s) > ), p Γ (.4) e ( s) and for choice µ = with sbstittion and, e.g., [3] [4] [5] [7] [9]) p = πq of the integration variable (see [] 977

7 ζ s π s d exp π, Re. s (.5) n= Γ + s ( s) = qq ( n q ) ( ( s) > ) Other choice of µ seems to be of lesser importance. Both representations (.4) and f t which (.5) are closely related to a Mellin transform ˆf ( s ) of a fnction together with its inversion is generally defined by (e.g., [5] [3] [33] [34] [35]) ˆ s f t f ˆ d tt f t, f λt f ( s), ( λ > ), s λ (.6) ˆ c i s d ˆ, ˆ s f s f t = st f s f ( s s ) t f ( t), iπ c i where c is an arbitrary real vale within the convergence strip of ˆf ( s ) in complex s -plane. The Mellin transform ˆf ( s ) of a fnction f ( t ) is closely related to the Forier transform ϕ ( y) of the fnction ϕ ( x) f ( e x ) by variable sbstittion t = e x and y is ζ s can be represented, =. Ths the Riemann zeta fnction sbstantially (i.e., p to factors depending on s ), as the Mellin transforms of the nt fnctions f ( t) = e = or of f n= t ( t) = e exp π nt, respectively. The n= kernels of the Mellin transform are the eigenfnctions of the differential operator t( t) to eigenvale s or, correspondingly, of the integral operator exp αt t of the mltiplication of the argment of a fnction by a factor e α (scaling of argment). Both representations (.4) and (.5) can be sed for the derivation of frther representations of the Riemann zeta fnction and for the analytic contination. The analytic contination of the Riemann zeta fnction can also be obtained sing the Eler-Maclarin smmation formla for the series in (.) (e.g., [5] [] [5]). Using the Poisson smmation formla, one can transform the representation (.5) of the Riemann zeta fnction to the following form s π s s q + q ζ ( s) = s( s) dq exp( π nq ). s (.7) q n=!( s ) This is known [] [3] [5] [7] [9] bt for convenience and de to the importance of this representation for or prpose we give a derivation in Appendix A. From (.7) which is now already tre for arbitrary complex s and, therefore, is an analytic contination of the representations (.) or (.5) we see that the Riemann zeta fnction satisfies a fnctional eqation for the transformation of the argment s s. In simplest form it appears by renormalizing this fnction via introdction of the xi ξ s defined by Riemann according to [] and to [5] [] fnction Riemann [] defines it more specially for argment s i t = + and writes it ( t) ξ with real t corres- ponding to or ξ + i t. Or definition agrees, e.g., with Eqation () in Section.8 on p. 6 of Edwards [5] and with [] and many others. 978

8 ξ ( s) s ( s )! ζ ( s), (.8) s π and we obtain for it the following representation converging in the whole complex plane of s (e.g., [] [4] [5] [7] [9]) s s q + q ξ ( s) = s( s) dq exp( π nq ), (.9) q with the normalization ξ = ξ = ζ ( ) =. (.) For s = the xi fnction and the zeta fnction possess the (likely transcendental) vales! 4 ξ = ζ , ζ = = (.) 4 π Contrary to the Riemann zeta fnction ζ ( s) the fnction ξ ( s) is an entire fnction. The only singlarity of ζ ( s) which is the simple pole at s =, is removed by mltiplication of ζ ( s) with s in the definition (.8) and the trivial zeros of ζ ( s) at s = n, ( n =,, ) are also removed by its mltiplication with s s! Γ + which possesses simple poles there. The fnctional eqation ξ s = ξ s, (.) from which follows for the n -th derivatives ( ) + ξ s = ξ ( s), ξ =, ( nm, =,,, ), (.3) n= n n n ( m ) and which expresses that ξ ( s) is a symmetric fnction with respect to s = as it is immediately seen from (.9) and as it was first derived by Riemann []. It can be easily ζ s converted into the following fnctional eqation for the Riemann zeta fnction s ( π) ζ ( s) = ζ ( s). s ( s ) *!cos π * Together with ξ( s) 6 ξ( s ) = we find by combination with (.) * * * * ( ) (.4) ξ s = ξ s = ξ s = ξ s, (.5) According to Havil [], (p. 93), already Eler correctly conjectred this relation for the zeta fnction ζ ( s) which is eqivalent to relation (.) for the fnction ξ ( s) bt cold not prove it. Only Riemann proved it first. 979

9 that combine in simple way, fnction vales for 4 points ( s, s, s *, s * ) of the complex plane. Relation (5) means that in contrast to the fnction ζ ( s) which is only real-valed on the real axis the fnction ξ ( s) becomes real-valed on the real * axis ( s = s = σ ) and on the imaginary axis ( s = it ). As a conseqence of absent zeros of the Riemann zeta fnction ( it) Re( s) ζ σ + for σ > together with the fnctional relation (4) follows that all nontrivial zeros of this fnction have to be within the strip σ and the Riemann hypothesis asserts that all zeros of the related xi fnction ξ ( s) are positioned on the so-called critical line s = + i, t ( < t < ). This is, in principle, well known. We se the fnctional Eqation (.) for a simplification of the notations in the following considerations and displace the imaginary axis of the complex variable s = σ + it from σ = to the vale σ = by introdcing the entire fnction Ξ ( z) of the complex variable z = x+ iy as follows Ξ( z) ξ + z, z = x+ iy = σ + i t = s, with the normalization (see (.) and (.)), Ξ ± = Ξ = ξ following from (.). Ths the fll relation of the Xi fnction ( z) zeta fnction ζ ( s) sing definition (.8) is + z z! 4 Ξ ( z) = ζ z. + z + 4 π (.6) (.7) Ξ to the Riemann (.8) We emphasize again that the argment displacement (.6) is made in the following only for convenience of notations and not for some more principal reason. The fnctional eqation (.) together with (.3) becomes ( n ) n ( n ) Ξ z =Ξ z, Ξ z = Ξ z, (.9) and taken together with the symmetry for the transition to complex conjgated variable * * * * ( ) Ξ z =Ξ z = Ξ z = Ξ z. (.) This means that the Xi fnction Ξ ( z) becomes real-valed on the imaginary axis z = iy which becomes the critical line in the new variable z ( y) ( y) ( ( y) ) ( ( y) ) * * Ξ i =Ξ i = Ξ i = Ξ i. (.) Frthermore, the fnction Ξ ( z) becomes a symmetrical fnction and a real-valed one on the real axis z = x ( x ) ( x ) ( ( x )) ( ( x )) * *. Ξ =Ξ = Ξ = Ξ (.) 98

10 A. Wünsche In contrast to this the Riemann zeta fnction ζ ( s ) the fnction is not a real-valed fnction on the critical line s= + it and is real-valed bt not symmetric on the real axis. This is represented in Figre. (calclated with Mathematica 6 sch as the frther figres too). We see that not all of the zeros of the real part Re ζ + it are also zeros of the imaginary part Im ζ + it and, vice versa, that not all of the zeros of the imaginary part are also zeros of the real part and ths genine zeros of the fnction ζ + it which are signified by grid lines. Between two zeros of the real part which are genine zeros of ζ + it lies in each case (exception first interval) an additional zero of the imaginary part, which almost coincides with a maximm of the real part. Figre. Real and imaginary part and absolte vale of Riemann zeta fnction on critical line. The position of the zeros of the whole fnction ζ + it on the critical line are shown by grid lines. One can see that not all zeros of the real part are also zeros of the imaginary part and vice versa. The figres are easily to generate by program Mathematica and are pblished in similar forms already in literatre. 98

11 Using (.9) and definition (.6) we find the following representation of Ξ ( z) z z q + q Ξ ( z) = z dq exp( π nq ). 4 (.3) q n= With the sbstittion of the integration variable A) representation (.3) is transformed to q = e (see also (.) in Appendix d ch e Ξ z z z exp( πn e ). 4 (.4) n= In Appendix A we show that (.4) can be represented as follows (see also Eqation (.) on p. 7 in [5] which possesses a similar principal form) ( z ) ( ) ( z ) Ξ = d Ω ch, (.5) with the following explicit form of the fnction Ω ( ) of the real variable Ω 4e πn e πn e 3 exp πn e >, ( < < ). (.6) n= The fnction Ω ( ) is symmetric Ω ( ) = +Ω( ) = Ω ( ), (.7) that means it is an even fnction althogh this is not immediately seen from representation (.6) 3. We prove this in Appendix A. De to this symmetry, formla (.5) can be also represented by Ξ z z = d ch z d ( ) e. Ω = Ω (.8) In the formlation of the right-hand side the fnction Ξ ( z) appears as analytic contination of the Forier transform of the fnction Ω ( ) written with imaginary argment z = iy or, more generally, with sbstittion z iz and complex z. From this follows as inversion of the integral transformation (.8) sing (.7) Ω iy ( ) = dy ( iy) e dy ( iy) cos ( y), π Ξ = π Ξ (.9) or de to symmetry of the integrand in analogy to (.5) Ω ( ) = dy ( iy) cos ( y), π Ξ (.3) where Ξ ( iy) is a real-valed fnction of the variable y on the imaginary axis * Ξ ( iy) = dω( ) cos ( y), ( Ξ ( iy) ) =Ξ( i y), de to (.5). (.3) A graphical representation of the fnction Ω ( ) and of its first derivatives 3 It was for s for the first time and was very srprising to meet a fnction where its symmetry was not easily seen from its explicit representation. However, if we sbstitte in (.6) and calclate and plot the part of Ω ( ) for with the obtained formla then we need mch more sm terms for the same accrateness than in case of calclation with (.6). 98

12 ( Ω ) ( ), ( n =,, 3) is given in Figre. The fnction ( ) Ω is monotonically de- ( creasing for < de to the non-positivity of its first derivative ) Ω( ) Ω ( ) which explicitly is (see also Appendix A) ( ) Ω = e πn e 8 πn e 3πn e + 5 exp πn e (.3) n=, <, ( Figre. Fnction Ω ( ) ) and its first derivative ( ) The fnction Ω ( ) is positive for ( Ω ) ( ) is negative for < < the fnction ( ) Ω (see (.5) and (.34)). < and since its first derivative Ω is mono- tonically decreasing on the real positive axis. It vanishes in infinity more rapidly than any exponential fnction with a polynomial in the exponent. 983

13 with one relative minimm at min.3766 = of depth ( ) Ω min = Moreover, it is very important for the following that de to presence of factors exp πn e Ω and ( ) in the sm terms in (.6) or in (.3) the fnctions ( ) ( ) Ω and all their higher derivatives are very rapidly decreasing for, more rapidly than any exponential fnction with a polynomial of in the argment. In this Ω is more comparable with fnctions of finite spport which sense the fnction vanish from a certain on than with any exponentially decreasing fnction. From (.7) follows immediately that the fnction Ω is antisymmetric that means it is an odd fnction. ( ) ( ) ( Ω = Ω ( ) = Ω( ), Ω ) =, (.33) It is known that smoothness and rapidness of decreasing in infinity of a fnction change their role in Forier transformations. As the Forier transform of the smooth (infinitely continosly differentiable) fnction Ω ( ) the Xi fnction on the critical line Ξ ( iy) is rapidly decreasing in infinity. Therefore it is not easy to represent the real-valed fnction Ξ ( iy) with its rapid oscillations nder the envelope of rapid decrease for increasing variable y graphically in a large region of this variable y. An appropriate real amplification envelope is seen from (.8) to be 4 π α ( y) = + iy! + 4y 4 fnction ζ + i t on the critical line i z = y. This is shown in Figre 3. The partial pictre for α ( y) Ξ ( iy) in Figre 3. with negative part folded p is identical with the absolte vale ζ + i t of the Riemann zeta fnction ζ ( s) on the imaginary axis s = + i t (forth partial pictre in Figre ). We now give a representation of the Xi fnction by the derivative of the Omega fnction. Using ch ( z) = sh ( z) one obtains from (.5) by partial integration z the following alternative representation of the fnction Ξ ( z) ( Ξ ( z) = d ) ( ) sh ( z), z Ω (.34) ( that de to antisymmetry of Ω ) ( ) and sh ( z ) with respect to can also be written Ξ ( ) ( ) z z = d sh z = d ( ) e. z Ω z Ω (.35) Figre gives a graphical representation of the fnction Ω ( ) and of its first ( derivative ) Ω Ω ( ) ( ) which de to rapid convergence of the sms is easily to which rises Ξ ( iy) to the level of the Riemann zeta 984

14 Figre 3. Xi Fnction ( iy) Ξ on the imaginary axis z = iy (corresponding to s = + iy ). The envelope over the oscillations of the real-valed fnction Ξ ( iy) decreases extremely rapidly with increase of the variable y in the shown intervals. This behavior makes it difficlt to represent this fnction graphically for large intervals of the variable y. By an enhancement factor which rises the amplitde to the level of the zeta fnction ζ ( s) we may see the oscillations nder the envelope (last partial pictre). A similar pictre one obtains for the modls of the Riemann zeta fnction ζ + iy only with or negative parts folded to the positive side of the ordinate, i.e. Ξ ( iy) = ζ + iy (see also Figre (last partial pictre)). The given vales for the zeros at ± iy n were first calclated by J.-P. Gram in 93 p to y 5 [5]. We emphasize here that the shown very rapid decrease of the Xi fnction at the beginning of y and for y ± is de to the very high smoothness of Ω ( ) for arbitrary. generate by compter. One can express Ξ ( z) also by higher derivatives n n Ω Ω ( ) ( ) of the Omega fnction Ω ( ) according to n ( m Ξ ( z) = d ) ( ) ch ( z m ) z Ω ( m+ = dω ) ( ) sh ( z),( m =,,, ), m+ z with the symmetries of the derivatives of the fnction Ω ( ) for ( m ) ( m ) ( m+ ) ( m+ ) ( m+ ( ) ) ( m ) (.36) Ω = +Ω, Ω = Ω, Ω =, =,,. (.37) 985

15 This can be seen by sccessive partial integrations in (.5) together with complete indction. The fnctions monotonic fnctions. ( n ) ( ) Ω in these integral transformations are for n not We mention yet another representation of the fnction Ξ ( z) formations the fnction ( z) ( ) n π e, d n π e d nd,. Using the trans- t n t = n = t (.38) Ξ according to (.8) with the explicit representation of the fnction Ω in (.6) can now be represented in the form z z 9 z 9 z Ξ ( z) = ( π n ) Γ +, πn + ( π n ) Γ, πn n= where Γ (, x) [36]) ( πn ) Γ + + Γ 4 4 z z 5 z 5 z 3 π n, πn π n, π n, (.39) α denotes the incomplete Gamma fnction defined by (e.g., [8] [] t α ( α x) t t ( α) γ ( α x) Γ, de Γ,. (.4) x However, we did not see a way to prove the Riemann hypothesis via the repre- sentation (.39). The Riemann hypothesis for the zeta fnction ζ ( s σ it) the hypothesis that all zeros of the related entire fnction ( z x iy) = + is now eqivalent to Ξ = + lie on the imaginary axis z = iy that means on the line to real part x = of z = x+ iy which becomes now the critical line. Since the zeta fnction ζ (s) does not possess zeros in the convergence region σ > of the Eler prodct (.) and de to symmetries (.7) and (.3) it is only necessary to prove that Ξ ( z) does not possess zeros within the strips x < and < x + to both sides of the imaginary axis z = iy where for symmetry the proof for one of these strips wold be already sfficient. However, we will go another way where the restriction to these strips does not play a role for the proof. 3. Application of Second Mean-Vale Theorem of Calcls to Xi Fnction After having accepted the basic integral representation (.5) of the entire fnction Ξ z according to ( z ) ( ) ( z ) Ξ d Ω ch, (3.) with the fnction Ω ( ) explicitly given in (.6) we concentrate s on its frther treatment. However, we do this not with this specialization for the real-valed fnction Ω ( ) bt with more general sppositions for it. Expressed by real part U( xy, ) and imaginary part V( xy, ) of Ξ ( z) 986

16 ( x y) U( xy) V( xy) U( xy) ( U( xy) ) V( xy) ( V( xy) ) * * Ξ + i, + i,,, =,,, =,, (3.) we find from (3.) (, ) = d Ω ch cos, (, ) = d Ω sh sin. (3.3) U x y x y V x y x y We sppose now as necessary reqirement for Ω ( ) and satisfied in the special case (.6) ( ) ( ) Ω >, <, Ω >. (3.4) Frthermore, ( z) is finite for arbitrary complex z and therefore that ( ) Ξ shold be an entire fnction that reqires that the integral (3.) Ω is rapidly decreasing in infinity, more precisely ( ) ( λ) Ω lim =, < λ <, exp (3.5) for arbitrary Ω shold be a nonsinglar fnction which is rapidly decreasing in infinity, more rapidly than any exponential fnction e λ with arbitrary λ >. Clearly, this is satisfied for the special fnction Ω ( ) in (.6). Or conjectre for a longer time was that all zeros of Ξ ( z) lie on the imaginary axis z = iy for a large class of fnctions Ω ( ) and that this is not very specific for the special fnction Ω ( ) given in (.6) bt is tre for a mch larger class. It seems that to this class belong all non-increasing fnctions Ω ( ), i.e sch fnctions for ( which holds Ω ) ( ) for its first derivative and which rapidly decrease in infinity. This means that they vanish more rapidly in infinity than any power fnctions n, ( n =,, ) (practically they vanish exponentially). However, for the convergence of the integral (3.) in the whole complex z -plane it is necessary that the fnctions have to decrease in infinity also more rapidly than any exponential fnction exp( λ) with arbitrary λ > expressed in (3.5). In particlar, to this class belong all rapidly decreasing fnctions Ω ( ) which vanish from a certain on and which may be called non-increasing finite fnctions (or fnctions with compact ( n spport). On the other side, continity of its derivatives Ω ) ( ),( n =,, ) is not reqired. The modified Bessel fnctions Iν ( z) normalized to the form of entire ν fnctions Iν ( z) for ν possess a representation of the form (3.) with z Ω λ. This means that the fnction fnctions Ω ( ) which vanish from on bt a nmber of derivatives of for the fnctions is not continos at = depending on the index ν. It is valable that here an independent proof of the property that all zeros of the modified Bessel fnctions Iν ( ) lie on the imaginary axis can be made sing their differential eqations via dality relations. We intend to present this in detail in a later work. Frthermore, to the considered class belong all monotonically decreasing fnctions with the described rapid decrease in infinity. The fine difference of the decreasing fnctions to the non-increasing fnctions Ω ( ) is that in first case the fnction 987

17 ( ) ( ) ( ) Ω cannot stay on the same level in a certain interval that means we have Ω < for all points > instead of ( ) ( ) Ω only. A fnction which de- creases not faster than e in infinity does not fall into this category as, for example, the fnction sech ( z) ch z shows. To apply the second mean-vale theorem it is necessary to restrict s to a class of Ω f which are non-increasing that means for which for all fnctions < in considered interval holds f a f f f b, a b, (3.6) or eqivalently in more compact form ( ) f, a b. (3.7) The monotonically decreasing fnctions in the interval a b, in particlar, belong to the class of non-increasing fnctions with the fine difference that here f a > f > f > f b >, a < < < b, (3.8) is satisfied. Ths smoothness of f ( ) for a < < b is not reqired. If frthermore g( ) is a continos fnction in the interval a b the second mean-vale theorem (often called theorem of Bonnet (867) or Gass-Bonnet theorem) states an eqivalence for the following integral on the left-hand side to the expression on the right-hand side according to (see some monographs abot Calcls or Real Analysis; we recommend the monographs of Corant [37] (Appendix to chap IV) and of Widder [38] who called it Weierstrass form of Bonnet s theorem (chap. 5, 4)) b d f ( ) g( ) = f ( a) dg( ) + f ( b) b d g( ), ( a b), (3.9) a a where is a certain vale within the interval bondaries a < b which as a rle we do not exactly know. It holds also for non-decreasing fnctions which inclde the monotonically increasing fnctions as special class in analogos way. The proof of the second mean-vale theorem is comparatively simple by applying a sbstittion in the (first) mean-vale theorem of integral calcls [37] [38]. Applied to or fnction f ( ) = Ω ( ) which in addition shold rapidly decrease in infinity according to (3.5) this means in connection with monotonic decrease that it has Ω > and therefore to be positively semi-definite if ( ( ) ) ( ) ( ) ( ) Ω Ω, Ω,, Ω, (3.) and the theorem (3.9) takes on the form dω ( ) g( ) = Ω d g( ), ( < ), (3.) where the extension to an pper bondary b in (3.9) for f ( ) = and in case of existence of the integral is nproblematic. If we insert in (3.9) for g( ) the fnction ch ( z ) which apart from the real variable depends in parametrical way on the complex variable z and is an analytic fnction of z we find that depends on this complex parameter also in an analytic 988

18 way as follows w ( z) sh ( w ( z) z) ( z) d ( ) ch ( z) dch ( z) ( ), w( z) ( x, y) i v( x, y), (3.) Ξ Ω =Ω =Ω = + z where w( x+ i z) = ( xy, ) + i v( xy, ) is an entire fnction with ( xy, ) its real and v ( xy, ) its imaginary part. The condition for zeros z is that sh ( w ( z) z ) vanishes that leads to ( ) w zz= xy, + i v xy, x+ iy = inπ, n=, ±, ±,, (3.3) or split in real and imaginary part xyx, v xy, y=, (3.4) for the real part and xy, y+ v xyx, = nπ, n=, ±, ±,, (3.5) for the imaginary part. The mlti-valedness of the mean-vale fnctions in the conditions (3.3) or (3.5) is an interesting phenomenon which is connected with the periodicity of the fnction g ( ) = ch ( z) on the imaginary axis z = iy in or application (3.) of the second mean-vale theorem (3.). To or knowledge this is p to now not well stdied. We come back to this in the next Sections 4 and, in particlar, Section 7 brings some illstrative clarity when we represent the mean-vale fnctions graphically. At present we will say only that we can choose an arbitrary n in (3.5) which provides s the whole spectrm of zeros z, z, on the pper half-plane and the corresponding spectrm of zeros z = z, z = z, on the lower half-plane of which as will be later seen lie all on the imaginary axis. Since in compter calclations the vales of π π the Arcs Sine fnction are provided in the region from to + it is convenient to choose n = bt all other vales of n in (3.5) lead to eqivalent reslts. One may represent the conditions (3.4) and (3.5) also in the following eqivalent form y x ( xy, ) = nπ, v ( xy, ) = nπ, (3.6) x + y x + y from which follows v ( xy, ) x ( ( xy, ) v ( xy, ))( x y) ( nπ ),. ( xy, ) y All these forms (3.4)-(3.7) are implicit eqations with two variables (, ) cannot be resolved with respect to one variable (e.g., in forms y y ( x) + + = = (3.7) xy which = k for each fixed n and branches k ) and do not provide immediately the necessary conditions for zeros in explicit form bt we can check that (3.6) satisfies the Cachy-Riemann eqations as a minimm reqirement ( xy, ) v( xy, ) ( xy, ) v( xy, ) =, =. x y y x (3.8) 989

19 We have to establish now closer relations between real and imaginary part ( xy, ) and v( xy ) of the complex mean-vale parameter w ( z = x+ y). The first step in, i preparation to this aim is the consideration of the derived conditions on the imaginary axis. 4. Specialization of Second Mean-Vale Theorem to Xi Fnction on Imaginary Axis By restriction to the real axis y = we find from (3.3) for the fnction Ξ ( z) Ξ ( x) = U( x, ), V ( x,) =, (4.) with the following two possible representations of U( x,) related by partial in- tegration U x x x x ( (,) = d Ω ch = d Ω ) sh >. (4.) Ω, Ω > Ω. Therefore, The ineqality U( x,) > follows according to the spposition ( ) from the non-negativity of the integrand that means from ( ) ch ( x) the case y = can be exclded from the beginning in the frther considerations for zeros of U( xy, ) and V( xy, ). We now restrict s to the imaginary axis x = and find from (3.3) for the fnction Ξ z ( iy) U(, y), V (, y). with the following two possible representations of U(, ) tegration Ξ = = (4.3) U y y y y From the obvios ineqality y related by partial in- ( (, ) = d Ω cos = d Ω ) sin. (4.4) cos ( y), together with the spposed positivity of ( ) sentation of U(, y ) in (4) the ineqality U( y) U ( ) + (4.5) Ω one derives from the first repre- Ω, +Ω, Ω =, d Ω. (4.6) In the same way by the ineqality ( y) representation of sin +, (4.7) one derives sing the non-positivity of Ω (see (3.)) together with the second U, y in (4.4) the ineqality ( ) Ω U, y y +Ω, Ω = dω. (4.8) which as it is easily seen does not depend on the sign of y. Therefore we have two non-negative parameters, the zeroth moment Ω and the vale Ω, which 99

20 y to an interior range both to (4.6) and to (4.8) at once. For mentioned prpose we now consider the restriction of the mean-vale parameter w ( z ) to the imaginary axis z = iy for which g ( ) = ch ( ( iy) ) = cos( y) is a realvaled fnction of y. For arbitrary fixed y we find by the second mean-vale theorem a parameter in the interval y < which natrally depends on the chosen vale y that means = (, y). The extension from the imaginary axis z = iy to the whole complex plane can be made then sing methods of complex analysis. We discss some formal approaches to this in Appendix B. Now we apply (3.) to the imaginary axis z = iy. The second mean-vale theorem (3.) on the imaginary axis z = iy (or x = ) takes on the form according to (4.6) and (4.8) restrict the range of vales of U(, ) (, y) dω ( ) ch ( ( iy) ) = dω cos y =Ω d cos y (4.9) sin ( (, y) y) =Ω( ), ( (, y), v(, y) = ). y As already said since the left-hand side is a real-valed fnction the right-hand side has also to be real-valed and the parameter fnction w ( iy ) is real-valed and therefore it can only be the real part (, y ) of the complex fnction w( z= x+ i y) = ( xy, ) + i v( xy, ) for x =. The second mean-vale theorem states that (, y ) lies between the minimal and maximal vales of the integration borders that is here between and and this means that (, y ) shold be positive. Here arises a problem which is connected with the periodicity of the fnction g ( ) = cos( y) as fnction of the variable for fixed variable y in the application of the mean-vale theorem. Let s first consider the special case y = in (4.9) which leads to ( y) sin d Ω =Ω d =Ω =Ω lim,, >. y y = From this relation follows (,) the contination to (, ) vales Ω and.78 and therefore (4.) > and it seems that all is correct also with y > for arbitrary y. One may even give the approximate Ω Ω.497 which, however, are not of importance for the later proofs. If we now start from (,) > and contine it continosly to (, y ) then we see that (, y ) goes monotonically to zero and approaches zero approximately at y = y 4.35 that is at the first zero of the fnction Ξ ( iy) on the positive imaginary axis and goes then first beyond zero and oscillates then with decreasing amplitde for increasing y arond the vale zero with intersecting it exactly at the zeros of Ξ ( iy). We try to illstrate this graphically in Section 7. All zeros lie then on the branch (, y) y = nπ with n =. That (, y ) goes beyond zero seems to contradict the content of the second mean-vale theorem according which (, y ) has to be positive in or 99

21 application. Here comes into play the mlti-valedness of the mean-vale fnction (, y ). For the zeros of sin ( (, y) y ) in (4.9) the relations (, y) y = nπ with different integers n are eqivalent and one may find to vales (, y ) < eqivalent crves ( n;, y ) with ( n;, y ) > and all these crves begin with ( n ;,) for y. However, we cannot contine (,) in continos way to only positive vales for (, y ). For y the ineqality (4.8) is stronger than (4.6) and characterizes the restric-, U, y y = Ω sin, y y follows tions of U( y ) and via the eqivalence ( ) from (4.8) (, ) Ω U y y n π (, y) y = arcsin n+ π, ( n =, ±, ±, ), (4.) where the choice of n determines a basis interval of the involved mlti-valed fnction arcsin ( z ) and the ineqality says that it is in every case possible to choose it from the same interval of length π. The zeros Ξ x+ iy on y of the Xi fnction the imaginary axis x = (critical line) are determined alone by the (mlti-valed) fnction (, y ) whereas v (, y ) vanishes atomatically on the imaginary axis in considered special case and does not add a second condition. Therefore, the zeros are the soltions of the conditions = ( = ± ± ) ( = ) k, y y nπ, n,,,, v, y. (4.) It is, in general, not possible to obtain the zeros y k on the critical line exactly from the mean-vale fnction (, y ) in (4.9) since generally we do not possess it explicitly. In special cases the fnction (, y ) can be calclated explicitly that is the case, for ν example, for all (modified) Bessel fnctions Iν ( z). The most simple case among z these is the case ν = when the corresponding fnction Ω ( ) is a step fnction Ω =Ω θ, (4.3) where θ ( x), x < =, x > is the Heaviside step fnction. In this case follows sh ( z) π z z =Ω Ξ z =Ω dch z =Ω =Ω I z, (4.4) where Ω = dω( ) is the area nder the fnction Ω ( ) =Ω θ ( ) (or the zeroth-order moment of this fnction. For the sqared modls of the fnction Ξ z we find ( z) ( ( z) ) ( ) sin ( ) * sh x + y ch x cos y Ξ Ξ = Ω = Ω, x + y ( x + y ) from which, in particlar, it is easy to see that this special fnction ( x iy) (4.5) Ξ + possesses 99

22 zeros only on the imaginary axis z = iy or x = and that they are determined by nπ y n = nπ, yn =, ( n= ±, ±, ). (4.6) The zeros on the imaginary axis are here eqidistant bt the soltion y = is absent since then also the denominators in (4.5) are vanishing. The parameter w ( z ) in the second mean-vale theorem is here a real constant in the whole complex plane w x+ i y = xy, + i v xy, =, xy, =, y =, v xy, = v, y =. (4.7) Practically, the second mean-vale theorem compares the reslt for an arbitrary fnction Ω ( ) nder the given restrictions with that for a step fnction Ω ( ) =Ω θ ( ) by preserving the vale Ω and making the parameter depending on z in the whole complex plane. Withot discssing now qantitative relations the formlae (4.7) sggest that v ( xy, ) will stay a small fnction compared with ( xy, ) in the neighborhood of the imaginary axis (i.e. for x y ) in a certain sense. We will see in next Section that the fnction (, y ) taking into accont v (, y ) = determines the fnctions ( xy, ) and v ( xy, ) and ths w ( z ) in the whole complex plane via the Cachy-Riemann eqations in an operational approach that means in an integrated form which we did not fond p to now in literatre. The general formal part is again delegated to an Appendix B. 5. Accomplishment of Proof for Zeros of Xi Fnctions on Imaginary Axis Alone In last Section we discssed the application of the second mean-vale theorem to the fnction Ξ ( z) on the imaginary axis z = iy. Eqations (3.4) and (3.5) or their eqivalent forms (3.6) or (3.7) are not yet sfficient to derive conclsions abot the position of the zeros on the imaginary axis in dependence on x. We have yet to derive more information abot the mean-vale fnctions w ( z ) which we obtain by relating the real-valed fnction ( xy, ) and v ( xy, ) to the fnction ( xy, ) on the imaginary axis taking into accont v (, y ) =. The general case of complex z can be obtained from the special case z = iy in (4.9) by application of the displacement operator exp ix to the fnction Ξ ( iy ) y according to Ξ ( x+ iy) = exp ix Ξ( iy) exp ix y y = exp ix dω ch iy exp ix y = exp ix Ω exp ix y y y sh, i i i = Ω. y sh ( i (, y) y) i ( ( y x) ( y x) ) i( y ix) (5.) 993

23 The fnction (, y ix) = w( x+ i y) = ( xy, ) + i v( xy, ) is related to ( y ) as follows ( xy, ) = cos x (, y), v( xy, ) = sin x (, y), y y or in more compact form w( x+ iy) = exp ix (, y) = (, y i x). y, (5.) (5.3) This is presented in Appendix B in more general form for additionally non- vanishing xy v (, y ) and arbitrary holomorphic fnctions. It means that we may obtain (, ) xy, by applying the operators cos x y and sin x, respectively, to y the fnction (, y ) on the imaginary axis (remind v (, y ) = vanishes there in or case). Clearly, Eqations (5.) are in agreement with the Cachy-Riemann eq- v v ations = and = as a minimal reqirement. x y y x and v We now write Ξ ( z) in the form eqivalent to (5.) ( x iy) Ξ + =Ω The denominator (( ( xy) + v( xy) )( x+ y) ) sh, i, i x+ iy (( ( xyx ) v( xy) y) + ( ( xy) y+ v( xyx ) )) sh,, i,, = Ω( ). x+ iy (5.4) x+ iy does not contribte to zeros. Since the Hyperbolic Sine possesses zeros only on the imaginary axis we see from (5.4) that we may expect zeros only for sch related variables ( xy, ) which satisfy the necessary condition of vanishing of its real part of the argment that leads as we already know to (see (3.4)) ( xyx, ) v( xy, ) y=. (5.5) The zeros with coordinates (, ) x y themselves can be fond then as the (in general non-degenerate) soltions of the following eqation (see (3.5)) k k xy, y+ v xyx, = nπ, n=, ±, ±,, (5.6) if these pairs ( xy, ) satisfy the necessary condition (5.5). Later we will see that it provides the whole spectrm of soltions for the zeros bt we can also obtain each ( xk, y k) separately from one branch n and wold they then denote by ( xn, y n). Ths we have first of all to look for sch pairs ( xy, ) which satisfy the condition (5.5) off the imaginary axis that is for x since we know already that these fnctions may possess zeros on the imaginary axis z = iy. Using (5.) we may represent the necessary condition (5.5) for the proof by the second mean-vale theorem in the form xcos x (, y) + ysin x (, y) =, y y (5.7) 994

24 and Eqation (5.6) which determines then the position of the zeros can be written with eqivalent vales n ycos x (, y) xsin x (, y) = nπ, ( n =, ±, ±, ). (5.8) y y We may represent Eqations (5.7) and (5.8) in a simpler form sing the following operational identities xcos x + ysin x = sin x y, ycos x xsin x = cos x y, y y y y y y (5.9) which are a specialization of the operational identities (B.) in Appendix B with wz = xy (, ) + i vxy (, ) z= x+ iy and therefore xy (, ) xvxy, (, ) y. If we mltiply (5.7) and (5.8) both by the fnction (, y ) then we may write (5.7) in the y, y =, y y) form (changing order and (5.8) in the form sin x ( (, y) y) =, y (5.) cos x ( (, y) y) = nπ, ( n =, ±, ±, ). (5.) y The left-hand side of these conditions possess the general form for the extension of a holomorphic fnction W( z x+ i y) = U( xy, ) + i V( xy, ) from the fnctions U(, y ) and V (, y ) on the imaginary axis to the whole complex plane in case of V (, y ) = and if we apply this to the fnction U(, y) = (, y) y. Eqations (5.) and (5.) possess now the most simple form, we fond, to accomplish the proof for the exclsive position of zeros on the imaginary axis. All information abot the zeros of the Xi fnction Ξ ( z) = U( xy, ) + i V( xy, ) for arbitrary x is now contained in the conditions (5.) and (5.) which we now discss. Since cos x is a nonsinglar operator we can mltiply both sides of eqation y (5.) by the inverse operator cos x and obtain y (, y ) y cos x n π π π, (,, ). y x = = + + y n = n n = ± (5.) This eqation is yet flly eqivalent to (5.) for arbitrary x bt it provides only the same possible soltions for the vales y of zeros as for zeros on the imaginary axis. This alone already sggests that it cannot be that zeros with x if they exist possess the same vales of y as the zeros on the imaginary axis. Bt in sch form the proof of the impossibility of zeros off the imaginary axis seemed to be not satisfactory and we present in the following some slightly different variants which go deeper into the details of the proof. 995