On the geometry of the Titchmarsh counterexample

Journal of Physics A: Mathematical and Theoretical PAPER • OPEN ACCESS On the geometry of the Titchmarsh counterexample To cite this article: W P Schleich et al 2022 J. Phys. A: Math. Theor. 55 484006 View the article online for updates and enhancements. You may also like Recovering a quantum graph spectrum from vertex data Jonathan Rohleder - A local inverse spectral theorem for Hamiltonian systems Matthias Langer and Harald Woracek - SOME PROBLEMS IN THE THEORY OF A STURM-LIOUVILLE EQUATION B M Levitan and I S Sargsyan - This content was downloaded from IP address 158.196.184.32 on 14/03/2023 at 10:30 Journal of Physics A: Mathematical and Theoretical J. Phys. A: Math. Theor. 55 (2022) 484006 (15pp) https://doi.org/10.1088/1751-8121/aca5d5 On the geometry of the Titchmarsh counterexample W P Schleich1,2,∗, I Tkácˇová3and H Maier4 1Institut für Quantenphysik and Center for Integrated Quantum Science and Technology (IQST), Universität Ulm, Albert-Einstein-Allee 11, Ulm, D-89081, Germany 2Institute for Quantum Science and Engineering (IQSE), and Texas A&M AgriLife Research and Hagler Institute for Advanced Study, Texas A&M University, College Station, TX 77843-4242, United States of America 3Department of Physics, Faculty of Electrical Engineering and Computer Science, VSB-Technical University of Ostrava, 17. Listopadu 2172/15, Ostrava—Poruba 70833, Czech Republic 4Institut für Zahlentheorie und Wahrscheinlichkeitstheorie, Universität Ulm, D-89081 Ulm, Germany E-mail: [email protected] Received 14 June 2022; revised 18 November 2022 Accepted for publication 24 November 2022 Published 9 December 2022 Abstract We study the lines of constant phase corresponding to the ratio formed by the building blocks of the Titchmarsh counterexample, that is by two Dirichlet L-functions whose characters are the complex conjugate of each other. This ratio on the critical line is sensitive to zeros off the critical line. Keywords: Riemann zeta function, Titchmarsh counterexample, lines of constant phase, lines of constant height (Some figures may appear in colour only in the online journal) 1. Introduction The verification of the Riemann Hypothesis, that is of the claim that the so-called non-trivial zeros of the Riemann zeta function ζare all located on the critical line in the complex plane defined by the real part 1/2 is a famous still unsolved problem in mathematics. In his seminal ∗Author to whom any correspondence should be addressed. Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1751-8121/22/484006+15$33.00 © 2022 The Author(s). Published by IOP Publishing Ltd Printed in the UK 1 J. Phys. A: Math. Theor. 55 (2022) 484006 W P Schleich et al book [1] Edward Charles Titchmarsh introduced a function ξTwhich shares many properties of the analytic continuation of ζ, that is the Riemann function ξ5and is known to have [2] zeros off the critical line. The appearance of these zeros has been explained [3] as a consequence of the universality [4] of the underlying Dirichlet L-functions. In the present article we point out a crucial difference between ξand ξTwhich manifests itself in the lines of constant phase [5] of the ratios formed by the elements of the corresponding analytic continuations. 1.1. Formulation of the problem For this purpose, we study a class of complex-valued functions Fcharacterized by their representation as a superposition F(s) = f(s) + f(1−s)(1) of a single complex-valued function fevaluated at s and at 1 −swhere s≡σ+iτ. Obviously, Fsatisfies the elementary functional equation F(s) = F(1−s).(2) A zero s0of Fappears when the ratio g(s)≡f(s) f(1−s)(3) assumes the value −1,that is g(s0) = −1.(4) This class includes but is not limited to an appropriately defined hyperbolic function c,the Riemann function ξand the Titchmarsh counterexample ξT.In the present article we show that for cand ξthe absolute values of the corresponding ratios gcand gRalong the critical line s=1/2+iτare unity. In contrast, for the Titchmarsh counterexample the ratio gTon the critical line is real, and assumes all values from −∞to +∞.This behavior originates from the fact that fTassociated with ξTdisplays zeros on the critical line, and the zeros of fT(s)and fT(1−s)are disjunct. As a result, poles and zeros appear in gT. In the case of two consecutive poles followed by two consecutive zeros, there must be two points, where the first derivative of gTvanishes. Here, two lines of constant phase leave symmetrically the critical line and unite again later. Provided the interval formed by the values of gTat the two points of vanishing derivative includes −1,there are two symmetrically located zeros of ξT. We suspect that this dramatically different behavior of gTversus gRon the critical line may provide us with yet another perspective on the Riemann Hypothesis. Indeed, we have already followed [6–8] an approach based on the lines of constant phase of ξand ξTrather than gRand gT. Moreover, we emphasize that the Riemann zeta function plays a central role not only in mathematics but also in physics. Three examples suffice to illustrate this point. (i) Indeed, the distribution of eigenvalues of Gaussian unitary ensembles of random matrices is similar [9] to that of the non-trivial zeros of the Riemann zeta function. (ii) There exists an intimate 5Throughout our article we refer to the product ξdefined by equation (11) which is free of the pole and the trivial zeros of the Riemann zeta function ζlocated at s=1 and the negative even integers, respectively, as the Riemann function. 2 J. Phys. A: Math. Theor. 55 (2022) 484006 W P Schleich et al connection [10] between the inverted harmonic oscillator and the Mellin transform of ξ;and (iii) the long-standing Polya–Hilbert Hypothesis of finding a Hamiltonian whose eigenvalues are given by the zeros of ξhas recently been verified [11]. 1.2. Outline Our article is organized as follows: In section 2we show that c, ξ, and ξTare elements of the class of functions defined by equation (1), and present explicit expressions for the corresponding building blocks fc,fRand fT.Then in section 3we demonstrate that for cand ξ the critical line is a line of constant height of gcand gRwith value unity. In sharp contrast, gTcorresponding to ξTis real along the critical line and thus a line of constant phase. This distinct difference between cand ξon one hand, and ξTon the other, is the deeper reason for our ability of identifying zeros of ξTlocated on and off the critical line by analyzing gTon it. Indeed, we demonstrate in section 4that gTassumes zeros and poles on the critical line and the geometry of the lines of constant phase of gTallows us to determine the location of these zeros. We conclude in section 5by providing a brief summary and an outlook. In order to keep our article self-contained we first briefly review in appendix Aproperties of Dirichlet L-functions Λcrucial for the main theme of our article. We then show that although in general their analytic continuation is not of the form, equation (1), the critical line is still a contour line of the corresponding ratio gΛwith value unity. Since the generalized Riemann Hypothesis states that all zeros of Λare also located on the critical line, this distinct property of gTcompared to gc,gRand gΛsupports the validity of the Riemann Hypothesis. 1.3. Dedication It is with great pleasure that we dedicate this article to Prof. Sir Michael Victor Berry on the occasion of his 80th birthday. We have chosen for our contribution the possibility of identifying zeros of the Titchmarsh counterexample off the critical line by the behavior of the ratio gTof the building blocks of ξTon the critical line. We are confident that this topic might find his interest since in a very stimulating discussion at the 600th Heraeus Seminar at Bad Honnef in 2015 we have learned from him that Ernest Oliver Tuck [12,13] has argued that his incompressibility function of ξon the critical axis line is sensitive to zeros off the critical line. Unfortunately, he was wrong as shown [14] by Michael Berry and Pragya Shukla. ‘Happy Birthday’ and many more happy and healthy years with fun in science! 2. Elements of class of functions Throughout our article we consider functions that satisfy the elementary functional equation, equation (2), and result from the superposition, equation (1) of a single function fevaluated at the two points sand 1 −s.In the present section we provide explicit expressions for fgiving rise to the hyperbolic cosine, the Riemann function and the Titchmarsh counterexample. Indeed, the most elementary example fc(s)≡1 2es−1/2(5) leads us by equation (1) to the function c(s)≡coshs−1 2.(6) 3 J. Phys. A: Math. Theor. 55 (2022) 484006 W P Schleich et al Moreover, for the choice fR(s)≡1 2s(s−1)γ(s) + 1 2(7) with the integral transform γ(s)≡ ∞ ˆ 1 dx ω(x)xs 2−1(8) of the Jacobi theta function ω(x)≡ ∞ X n=1 e−πn2x(9) we arrive at the familiar analytic continuation ξ(s) = 1 2s(s−1)[γ(s) + γ(1−s)] + 1 2(10) of the Riemann function ξ(s)≡π−s/2(s−1)Γs 2+1ζ(s),(11) where Γand ζdenote the Gamma and the Riemann zeta function, respectively. Finally, we consider the case fT(s)≡1 2 cosθe−iθΛ(s,χ1)(12) with the Dirichlet L-function Λ = Λ(s,χ1)of complex-valued character χ1mod 5 where the parameter θis defined by the condition tan(2θ) = √5−1 2. For a general introduction into and an overview over Dirichlet L-functions, we refer to [15, 16]. However, in appendix Awe briefly summarize properties of Λrelevant to the present discussion. Indeed, from equation (A10) we recall the functional equation Λ(1−s,χ1) = e2iθΛ(s,χ2)(13) and thus arrive at the expression fT(1−s) = 1 2cosθeiθΛ(s,χ2),(14) leading us by the superposition, equation (1), of fT(s)and fT(1−s)to the Titchmarsh counterexample ξT(s)≡1 2 cosθe−iθΛ(s,χ1) + eiθΛ(s,χ2).(15) Hence, the class of functions given by equation (1) includes all three examples. 3. Des Pudels Kern The crucial difference between the functions cand ξRon the one hand, and ξTon the other, stands out most clearly when we consider for each of them the value f(s∗)where the star 4 J. Phys. A: Math. Theor. 55 (2022) 484006 W P Schleich et al indicates the complex conjugate. This operation tests the symmetry of fwith respect to the real axis. Indeed, we find for cand ξRthe symmetry relations fc(s∗) = f∗ c(s)(16) and fR(s∗) = f∗ R(s).(17) However, the dependence of fTon the complex-valued character χ1, and the presence of the phase factor exp(−iθ),prevent a similar relation for the building block fTof the Titchmarsh counterexample, that is fT(s∗)=f∗ T(s).(18) In this section we first verify the identities, equations (16) and (17) as well as show the breakdown of this symmetry for fTas expressed by equation (18). We then demonstrate that this distinct difference implies that the critical line is a line of constant height for the ratios gc and gRformed by fcand fR, but is a line of constant phase for gTdefined by fT. 3.1. Confirmation and breakdown of symmetry with respect to real axis We start by noting that the definition, equation (5), of fcimmediately implies the symmetry relation, equation (16). A slightly more complicated argument verifies the corresponding property, equation (17), for ξR. Indeed, since the integration variable xin γgiven by equation (8) is real the Jacobi theta function ωdefined by equation (9) is real as well, and with the identity xs∗/2=exps∗ 2lnx=hexps 2lnxi∗ (19) we find γ(s∗) = γ(s)∗.(20) Moreover, the polynomial s(s−1)satisfies the relation s∗(s∗−1)=[s(s−1)]∗which with the definition, equation (7), of fRleads us to equation (17). Finally we address the case of ξTwhere according to the definition, equation (12), we find fT(s∗) = 1 2 cosθe−iθΛ(s∗,χ1),(21) which due to the dependence of Λon χ1takes the form fT(s∗) = 1 2 cosθe−iθ[Λ(s,χ∗ 1)]∗.(22) When we recall the symmetry relation χ∗ 1=χ2, for the character χ1we obtain the expression fT(s∗) = 1 2 cosθe−iθ[Λ(s,χ2)]∗(23) which is obviously not identical to f∗ T.Two features of fTprevent this identity: (i) the presence of the phase factor exp(−iθ)in front of the Dirichlet function Λ, and (ii) the dependence of Λ on χ1which leads to the emergence of χ2rather than χ1. 5 J. Phys. A: Math. Theor. 55 (2022) 484006 W P Schleich et al 3.2. The critical line: line of constant height of ratios gcand gR The symmetry relations, equations (16) and (17) for fcand fRhave an immediate consequence on the behavior of the corresponding ratios gcand gRon the critical line s=1/2+iτ. Indeed, for cand ξRwe obtain the representations gc1 2+iτ=fc1 2+iτ fc1 2−iτ=fc1 2+iτ fc1 2+iτ∗=ei2φc(τ)(24) and analogously gR1 2+iτ=ei2φR(τ)(25) where φc=φc(τ)and φR=φR(τ)are the phases of fcand fRalong the critical line. Hence, we arrive at the identity  gc1 2+iτ = gR1 2+iτ =1 (26) which indicates that along the critical line, gcand gRhave lines of constant height with value unity. According to equation (4) zeros of cand ξRoccur on the critical line for the imaginary parts τ(k) cand τ(k) Rwhere the phases φcand φRof gcand gRassume odd integer multiples of π, leading us to the condition φcτ(k) c= (2k+1)π 2(27) and φRτ(k) R= (2k+1)π 2.(28) Here kis an integer. Since in the case of cthe phase φcis just τ, we obtain the familiar explicit formula τ(k) c= (2k+1)π 2(29) for the imaginary parts τ(k) cof the zeros of con the critical line. Unfortunately, due to the more complicated expression for fRgiven by equations (7)–(9) no analytic expression for φR(τ)is known. Despite this complication, the zeros of ξRon the critical line follow from a condition identical to that of c. Indeed, they are located where the phase lines of gRwith an odd integer multiple of πcross the critical line. 3.3. The critical line: line of constant phase of ratio gT We now turn to the case of ξTand study the ratio gT1 2+iτ=e−2iθΛ1 2+iτ,χ1 Λ1 2+iτ,χ2(30) following from the definition, equation (3), of gwith equations (12) and (14). We note that on the critical line the functional equation, equation (A9), of Λ(s,χ1)reads Λ1 2+iτ,χ1=e2iθΛ1 2+iτ,χ1∗ (31) 6 J. Phys. A: Math. Theor. 55 (2022) 484006 W P Schleich et al and enforces the representation Λ1 2+iτ,χ1=eiθλ1(τ),(32) where λ1=λ1(τ)is a real function which is not necessarily positive. Similarly, we find from the functional equation, equation (A10), the expression Λ1 2+iτ,χ2=e−iθλ2(τ),(33) where λ2is a real function which is not necessarily positive. Since χ1=χ2, the two functions λ1and λ2are different. When we substitute the representations equations (32) and (33) into the expression, equation (30), of gT,we arrive at the relation gT1 2+iτ=λ1(τ) λ2(τ)≡λ(τ).(34) Hence, on the critical line gTis real, and according to equation (4) a zero s0=1 2+iτ0of ξTarises when gT1 2+iτ0=−1.(35) A comparison between the behavior of the ratios gc,gRand gTon the critical line expressed by equations (24), (25) and (34), brings out the importance of the symmetry with respect to the real axis. Indeed, for functions such as cor ξwhich enjoy this symmetry, the critical line is a line of constant height of gwith |g|=1.In this case, the phase of gcan assume any value as τincreases, and a zero of cor ξarises for an odd integer multiple of π. In contrast, a violation of this symmetry as displayed by ξT,leads to a situation where the ratio gTis real along the critical line. When we allow for zeros or poles of gT,the critical line is a line of piecewise constant phase with φTbeing an integer multiple of π. Zeros of ξTon the critical line arise at τ−values where gTis −1 as indicated by equation (35). 4. Zeros off the critical line So far we have concentrated on the mechanism for the appearance of zeros of F on the critical line. Here the ratio ghas played a central role. We now show that the behavior of g on the critical line, even allows us to identify zeros of Fthat are located off the critical line. We demonstrate this property using ξTwhich has such zeros. For this purpose we first recall that the Dirichlet L-functions Λ1= Λ1(s)≡Λ(s,χ1)and Λ2= Λ2(s)≡Λ(s,χ2)have simple zeros [3,6] on the critical line6as exemplified by figure 1. Hence, at a zero of Λ2the ratio gThas a simple pole, and at a zero of Λ1a zero. Since poles are sources and zeros are sinks of phase lines [6] we find a flow pattern of gTillustrated in figure 2. The influence of the locations of the zeros and poles of gTon the zeros of ξTshown on the left of figure 2stands out most clearly when we consider for increasing τthe sequence of two zeros and two poles of gTlocated on the critical line as depicted in the middle of figure 2. On the right we present gTon the critical line where according to equation (34)gTis real. 6We emphasize that for our argument the still unverified generalized Riemann Hypothesis, that is the claim that all zeros of Λare on the critical line, is not of importance. 7 J. Phys. A: Math. Theor. 55 (2022) 484006 W P Schleich et al Figure 1. Lines of constant phase (grey lines) for the Dirichlet L-functions Λ1(s)≡ Λ(s,χ1)(left) and Λ2≡Λ(s,χ2)(middle) corresponding to the characters χ1and χ2≡ χ∗ 1defined by equation (A6), compared and contrasted to the ones of the Titchmarsh counterexample ξT(right) in identical domains of the complex plane s≡σ+iτ. The phase shifts of +2θand −2θin Λ1and Λ2with respect to the critical line σ=1/2 are a result of the corresponding phases of the normalized Gauss sums, equation (A5), and the functional equations, equations (A9) and (A10). In contrast, ξTwhich is determined by a superposition, equation (15), of Λ1and Λ2such that it satisfies the functional equation, equation (2) does not display a phase shift but enjoys an antisymmetry of its phase. Due to its simple pole, gThas to either increase from the second zero in the middle of figure 2 to plus infinity and increase from minus infinity after the pole, or decrease from the second zero to minus infinity and decay from plus infinity after the pole. Only in the second scenario, the condition, equation (35) for a zero of ξTis satisfied for a value of τbefore the pole as shown on the right of figure 2. Moreover, we note that after the second pole gTincreases from minus infinity through the top zero of gT.Hence, there must be another zero of ξTbetween this pole and this zero of gT. So far we have explained the emergence of a zero of ξTon the critical line. We now discuss the mechanism underlying the appearance of a zero off the critical line. For this purpose we consider a situation depicted in the middle of figure 3where Λ1has two consecutive zeros which are not separated by a zero of Λ2. As a consequence, gTdisplays two adjacent simple zeros with a zero of the first derivative g′ Tin between. This point τ′ 0on the critical line marked by a green triangle in the middle of figure 3is the starting point of two lines of constant phase shown in green that move away symmetrically from their point of birth into the complex plane, and return to the critical line at another point ˜τ′ 0on the critical line, 8 J. Phys. A: Math. Theor. 55 (2022) 484006 W P Schleich et al A.2. Critical line is line of constant height of gΛ Finally, we turn to the ratio gΛ(s,χ)≡e−iβ(χ)γ(s,χ) γ(1−s,χ∗) defined in analogy to equation (3) and following from the analytic continuation, equation (A2), of Λwhich on the critical line reads gΛ1 2+iτ,χ=e−iβ(χ)γ1 2+iτ,χ γ1 2+iτ,χ∗. As a result, we arrive at the expression gΛ1 2+iτ,χ=e−iβ(χ)e2iδΛ(τ;χ)(A11) where δΛ=δΛ(τ;χ)is the phase of the integral transform γ1 2+iτ,χof the generalized theta function ω(x,χ)on the critical line. Hence, the critical line is indeed a contour line of gΛwith gΛ1 2+iτ,χ=1. ORCID iD W P Schleich https://orcid.org/0000-0002-9693-8882 References [1] Titchmarsh E C 1951 The Theory of the Riemann Zeta-Function (Oxford: Clarendon) [2] Spira R 1994 Some zeros of the Titchmarsh counterexample Math. Comput. 63 747–8 [3] Karatsuba A A and Voronin S M 1992 The Riemann Zeta-Function (Berlin: de Gruyter) [4] Voronin S M 1975 Theorem on the “universality” of the Riemann zeta-function Izv. Akad. Nauk SSSR Ser. Mat. 39 475–86 Voronin S M Math. USSR Izv. 9443–5 (Engl. transl.) [5] Schleich W P, Tkácˇová I and Happ L 2022 Insights into complex functions (Lecture Notes in Physics vol 1000) (Heidelberg: Springer) [6] Neuberger J W, Feiler C, Maier H and Schleich W P 2014 Newton flow of the Riemann zeta function: separatrices control the appearance of zeros New J. 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