Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function
aa r X i v : . [ m a t h . N T ] F e b Contemporary Mathematics
Truncated Infinitesimal Shifts, Spectral Operators andQuantized Universality of the Riemann Zeta Function
Hafedh Herichi and Michel L. Lapidus
A Christophe Soul´e, avec une profonde amiti´e et admiration, `a l’occasion de ses 60 ans
Abstract.
Nous rappelons quelques unes des principales propri´et´es d’universalit´ede la fonction zˆeta de Riemann ζ ( s ). De plus, nous expliquons comment obtenirune quantification naturelle du th´eor`eme d’universalit´e de Voronin (et deses g´en´eralizations). Notre travail est bas´e sur la th´eorie des cordes fractalesd´evelopp´ee par le deuxi´eme auteur et M. van Frankenhuijsen dans [ La-vF4 ].Nous utilisons ´egalement la th´eorie d´evelopp´ee dans [
HerLa1–3 ] par les au-teurs de cet article pour ´etudier de fa¸con rigoreuse l’op´erateur spectral (quirelie la g´eom´etrie et le spectre des cordes fractales g´en´eraliz´ees). Cet op´erateurspectral est represent´ee comme le compos´e de la fonction zˆeta de Riemanndu ‘shift infinitesimal’ ∂ : a = ζ ( ∂ ). Dans le processus du quantification duth´eor`eme d’universalit´e de la fonction zˆeta de Riemann, le rˆole jou´e par lavariable s (au sens classique du th´eor´eme d’universalit´e) (dans le th´eor`eme clas-sique d’universalit´e) est jou´e par la famille des shifts infinit´esimaux tronqu´esafin d’´etudier l’op´erateur spectral en lien avec la reformulation spectrale del’hypoth´ese de Riemann vue comme un probl`eme spectral inverse pour lescordes fractales. Ce dernier r´esultat fournit une version op´eratorielle de la re-formulation spectrale obtenue par le second auteur et H. Maier dans [ LaMa2 ].Notre pr´esent travail au long terme, ainsi que [
La5, La6 ], a en partie pourbut d’obtenir une quantification naturelle de divers aspects de la th´eorie ana-lytiques des nombres et de la g´eom´etrie arithm´etiques.2010
Mathematics Subject Classification. Primary
Secondary
Key words and phrases.
Riemann zeta function, Riemann zeros, Riemann hypothesis, spec-tral reformulations, fractal strings, complex dimensions, explicit formulas, geometric and spectralzeta functions, geometric and spectral counting functions, inverse spectral problems, infinitesimalshift, truncated infinitesimal shifts, spectral operator, truncated spectral operators, universalityof the Riemann zeta function, universality of the spectral operator.The work of M. L. Lapidus was partially supported by the US National Science Foundationunder the research grant DMS-1107750, as well as by the Institut des Hautes Etudes Scientifiques(IHES) where the second author was a visiting professor in the Spring of 2012 while part of thiswork was completed. c (cid:13) Contents
1. Introduction 22. Universality of the Riemann Zeta Function 32.1. The Riemann zeta function ζ ( s ) 42.2. Voronin’s original universality theorem for ζ ( s ) 52.3. Some applications of the universality of ζ ( s ) 63. Generalized Fractal Strings and the Spectral Operator a = ζ ( ∂ c ) 93.1. Generalized fractal strings and explicit formulas 93.2. The spectral operator and the infinitesimal shifts ∂ c of the real line 123.3. Properties of the infinitesimal shifts ∂ c a ( T ) c = ζ ( ∂ ( T ) c ) 184.1. The truncated infinitesimal shifts ∂ ( T ) c and their properties 184.2. The truncated spectral operators and their spectra 195. Truncated Infinitesimal Shifts and Quantum Universality of ζ ( s ) 215.1. An operator-valued extension of Voronin’s theorem 215.2. A more general operator-valued extension of Voronin’s theorem 236. Concluding Comments 257. Appendix A: On the Origins of Universality 268. Appendix B: On some Extensions of Voronin’s Theorem 278.1. A first extension of Voronin’s original theorem 278.2. Further extensions to L-functions 289. Appendix C: Almost Periodicity and the Riemann Hypothesis 30References 31
1. Introduction
The universality of the Riemann zeta function states that any non-vanishinganalytic function can be approximated uniformly by certain purely imaginary shiftsof the zeta function in the right half of the critical strip [
Vor2 ]. It was discoveredby S. M. Voronin in 1975. Several improvements of Voronin’s theorem are given in[
Bag1, Lau1, Rei1, Rei2 ]. Further extensions of this theorem to other classesof zeta functions can be found in [
Emi, Lau2, LauMa1, LauMa2, LauMaSt,LauSlez, LauSt, St1 ]. In the first part of the present paper, as well as in severalappendices, we survey some of these results and discuss their significance for theRiemann zeta function ζ and other L -functions.In the second part of the paper, we focus on the Riemann zeta function ζ ( s )and its operator (or ‘quantum’) analog, and propose a quantum (or operator-valued)version of the universality theorem and some of its extensions. More specifically, intheir development of the theory of complex dimensions in fractal geometry andnumber theory, the spectral operator was introduced heuristically by the secondauthor and M. van Frankenhuijsen as a map that sends the geometry of fractalstrings to their spectra [ La-vF3, La-vF4 ]. A detailed, rigorous functional analyticstudy of this map was provided by the authors in [
HerLa1 ]. RUNCATED INFINITESIMAL SHIFTS AND QUANTIZED UNIVERSALITY OF ζ ( s ) 3 In this paper, we discuss the properties of the family of ‘ truncated infinitesi-mal shifts ’ of the real line and of the associated ‘ truncated spectral operators ’. (Seealso [
HerLa2, HerLa3, HerLa4 ].) These truncated operators were introduced in[
HerLa1 ] to study the invertibility of the spectral operator and obtain an operator-theoretic version of the spectral reformulation of the Riemann hypothesis (obtainedby the second author and H. Maier in [
LaMa1, LaMa2 ]) as an inverse spectralproblem for fractal strings. In particular, we show that a = ζ ( ∂ ), where ∂ is the‘infinitesimal shift’ (the generator of the translations on the real line), viewed asan unbounded normal operator acting on a suitable scale of Hilbert spaces indexedby the (Minkowski) fractal dimension of the underlying (generalized) fractal strings.Moreover, using tools from the functional calculus and our detailed study ofthe spectra of the operators involved, we show that any non-vanishing holomorphicfunction of the truncated infinitesimal shifts can be approximated by imaginarytranslates of the truncated spectral operators. This latter result provides a ‘natu-ral quantization’ of Voronin’s theorem (and its extensions) about the universalityof the Riemann zeta function. We conclude that, in some sense, arbitrarily smallscaled copies of the spectral operator are encoded within itself. Therefore, we de-duce that the spectral operator can emulate any type of complex behavior and thatit is ‘chaotic’. In the long term, the theory developed in the present paper and in[ HerLa1, HerLa2, HerLa3, HerLa4 ], along with the work in [
La5, La6 ], isaimed in part at providing a natural quantization of various aspects of analytic(and algebraic) number theory and arithmetic geometry.The rest of this paper is organized as follows: in §
2, we discuss the classicaluniversality property of the Riemann zeta function and some of its applications. In §
3, we briefly review the theory of (generalized) fractal strings and the associatedcomplex dimensions and explicit formulas, as developed in [
La-vF4 ]. We also recallthe heuristic definition of the spectral operator introduced in [
La-vF3 , La-vF4 ]. In §
4, we develop the rigorous functional analytic framework of [
HerLa1 ]; we defineand study, in particular, the infinitesimal shift ∂ and the spectral operator a , alongwith their truncated versions, ∂ ( T ) and a ( T ) (for T > §
5, we provide ourquantization (or operator-valued version) of Voronin’s theorem for the universal-ity of ζ ( s ), along with its natural generalizations in this context. It is noteworthythat in this ‘quantization process’, the complex variable s is replaced not by ∂ (the infinitesimal shift), as one might reasonably expect, but by the family of trun-cated infinitesimal shifts { ∂ ( T ) } T > . In §
6, we propose several possible directionsfor future research in this area. Finally, in three appendices, we provide some addi-tional information and references about the origins of universality, as well as aboutthe extensions of Voronin’s universality theorem to other arithmetic zeta functions(including the Dirichlet L -functions and the L -functions associated with certainmodular forms).
2. Universality of the Riemann Zeta Function
In this section, we recall some of the basic properties of the Riemann zetafunction (in § § ζ ( s ) (among allnon-vanishing holomorphic functions). We also briefly discuss (in § HAFEDH HERICHI AND MICHEL L. LAPIDUS mathematical and physical applications of universality. ζ ( s ) . The Riemann zeta function isdefined as the complex-valued function ζ ( s ) = ∞ X n =1 n − s , for Re ( s ) > . (2.1.1)In 1737, Euler showed that this Dirichlet series can be expressed in terms ofan infinite product over the set P of all the prime numbers: ζ ( s ) = ∞ X n =1 n − s = Y p ∈P − p − s , for Re ( s ) > . (2.1.2)Note that Equation (2.1.2) shows that the Riemann zeta function carries informa-tion about the primes, which are encoded in its Euler product.In 1858, Riemann showed in [ Rie ] that this function has a meromorphiccontinuation to all of C with a single (and simple) pole at s = 1, which satisfies the functional equation ξ ( s ) = ξ (1 − s ) , s ∈ C , (2.1.3)where ξ ( s ) := π − s Γ( s ζ ( s ) (2.1.4)is the completed (or global ) Riemann zeta function (Here, Γ denotes the classicgamma function.) Note that the trivial zeros of ζ ( s ) at s = − n for n = 1 , , , ..., correspond to the poles of the gamma function Γ( s ). Riemann also conjectured thatthe nontrivial (or critical ) zeros of ζ ( s ) (i.e., the zeros of ζ ( s ) which are located inthe critical strip 0 < Re ( s ) <
1) all lie on the critical line Re ( s ) = . This famousconjecture is known as the Riemann hypothesis .It is well known that the Euler product in Equation (2.1.2) converges ab-solutely to ζ ( s ) for Re ( s ) > Re ( s ) >
1. We note that the Euler product (or an appropriate substitutethereof) can be useful even in the critical strip { < Re ( s ) < } where it does notconverge. For example, it turns out that a suitable truncated version of this Eulerproduct, namely, ζ N ( s ) = Y p ≤ N (cid:0) − p − s (cid:1) − (2.1.5)(possibly suitably randomized), played a key role in Voronin’s proof of the uni-versality of the Riemann zeta function. This idea was due to Harald Bohr’s earlierwork on the density of the sets consisting of the ranges of ζ ( s ) on the vertical lines L s = { s ∈ C : Re ( s ) = c } , with < c <
1. Although it is a known fact that theRiemann zeta function’s Euler product (see the right-hand side of Equation (2.1.2))does not converge to ζ ( s ) inside the critical strip (in particular, inside the right-hand side of the critical strip < Re ( s ) < In 1740, Euler initiated the study of the Dirichlet series given in Equation (2.1.1) for thespecial case when the complex number s is a positive integer. Later on, his work was extended byChebychev to Re ( s ) >
1, where s ∈ C . RUNCATED INFINITESIMAL SHIFTS AND QUANTIZED UNIVERSALITY OF ζ ( s ) 5 truncated version of this Euler product can be used to approximate ζ ( s ) on theright-hand side of the critical strip, i.e., on the half-plane { < Re ( s ) < } (see[ Vor2 ]).The Riemann zeta function has a number of applications in the mathematicaland physical sciences (also, in biology and economics). For instance, in analyticnumber theory, the identity given in Equation (2.1.2) can be used to show thatthere are infinitely many primes among the integers. The Riemann zeta functionplays a key role in describing the distribution of the prime numbers. It also appearsin applied statistics (in the Zipf–Mandelbrot law), as well as in quantum field theory(in the calculation of the Casimir effect). In fractal geometry (for instance in thetheory of complex dimensions), the Riemann zeta function naturally occurs as amultiplicative factor in the formula relating the geometry and spectra of fractalstrings via their geometric and spectral zeta functions (see [
La2, La3, LaMa2,La-vF4 ]). The Riemann zeta function has several other interesting applications inphysics. (See, e.g., [
Tit, Edw, Ing, Ivi, Pat, KarVo, Ser, La-vF4, La5 ] for amore detailed discussion of the theory of the Riemann zeta function and some ofits applications; see also Riemann’s 1858 original paper [
Rie ].) It will be shown inthe next section that this function is also the first explicit universal object thatwas discovered in the mathematical sciences. ζ ( s ) . The universal-ity of the Riemann zeta function was established by S. M. Voronin in 1975. Thisimportant property of the Riemann zeta function states that any non-vanishing(i.e., nowhere vanishing) analytic function can be uniformly approximated by cer-tain purely imaginary shifts of the zeta function in the right half of the critical strip[
Vor2 ]. The following is Voronin’s original universality theorem: Theorem . Let < r < . Suppose that g ( s ) is a non-vanishing continousfunction on the disk D = { s ∈ C : | s | ≤ r } , which is analytic ( i.e., holomorphic ) in the interior ˚D = { s ∈ C : | s | < r } of this disk. Then, for any ǫ > , there exists τ ≥ such that J ( τ ) := sup | s |≤ r | g ( s ) − ζ ( s + 34 + iτ ) | < ǫ. (2.2.1) Moreover, the set of such τ ’s is infinite. In fact, it has a positive lower density;i.e., lim inf T →∞ T vol ( { τ ∈ [0 , T ] : J ( τ ) < ǫ } ) > , (2.2.2) where vol denotes the Lebesgue measure on R . The Riemann zeta function is said to be universal since suitable approximatetranslates (or imaginary shifts) of this function uniformly approximate any analytictarget function satisfying the hypothesis given above in Theorem 2.1. The domainof the uniform approximation of the admissible target function is called the stripof universality . See § ζ ( s ) and see also Appendix Aabout the origins of universality in the mathematical literature. We refer to Appendix B for several extensions of Voronin’s universality theorem to otherelements of the Selberg class of zeta functions.
HAFEDH HERICHI AND MICHEL L. LAPIDUS
The strongest version of Voronin’s theorem (the extended Voronin theorem)is due to Reich and Bagchi (see [
KarVo, Lau1, Lau2, St1 ]); it is given in thefollowing result:
Theorem . Let K be any compact subset of the right critical strip {
KarVo, Lau1, St1 ] for addi-tional information concerning the universality properties of the Riemann zeta func-tion and of other arithmetic zeta functions. (See also Appendix B below, providedin § ζ ( s ) . The universalitytheorem (Theorem 2.2) for the Riemann zeta function has several interesting ap-plications, which are related to functional independence, the critical zeros of theRiemann zeta function and therefore, to the Riemann hypothesis (this latter resultwas obtained in the work of Bagchi [
Bag1, Bag2 ]), the approximation of certaintarget functions by Taylor polynomials of zeta functions [
GauCl ], and also to pathintegrals in quantum theory (see [
BitKuRen ]).
RUNCATED INFINITESIMAL SHIFTS AND QUANTIZED UNIVERSALITY OF ζ ( s ) 7 Extension of the Bohr–Courant density theorem
The universality theorem (Theorem 2.2) for the Riemann zeta function provides anextension (due to Voronin in [
Vor1 ]) of the Bohr–Courant classical result [
BohCou ]about the density of the range of ζ ( s ) on an arbitrary vertical line contained in theright critical strip { s ∈ C : < Re ( s ) < } . The following result (also due toVoronin in [ Vor1 ]) can either be deduced from the universality theorem or proveddirectly (as was done originally in [
Vor1 ]):
Theorem . Let x ∈ ( , be fixed. ( Here, Re ( s ) = x and Im ( s ) = y . ) Then,for any integer n ≥ , the sets (cid:8)(cid:0) (log ζ ( x + iy )) , (log ζ ( x + iy )) ′ , ..., (log ζ ( x + iy )) ( n − (cid:1) : y ∈ R (cid:9) (2.3.1) and (cid:8) ζ ( x + iy ) , ( ζ ( x + iy )) ′ , ..., ( ζ ( x + iy )) ( n − (cid:1) : y ∈ R (cid:9) (2.3.2) are dense in C n . Functional independence and hypertranscendence of ζ ( s )Furthermore, the universality of ζ ( s ) implies functional independence. In 1887,H¨older proved the functional independence of the gamma function, i.e., that thegamma function Γ( s ) does not satisfy any algebraic differential equation. In otherwords, for any integer n ≥
1, there exists no non-trivial polynomial P such that P (Γ( s ) , Γ ′ ( s ) , ..., Γ ( n − ( s )) = 0 . (2.3.3)In 1900, and motivated by this fact, Hilbert suggested at the InternationalCongress for Mathematicians (ICM) in Paris that the algebraic differential indepen-dence of the Riemann zeta function can be proved using H¨older’s above result andalso the functional equation for ζ ( s ). In 1973, and using a ‘suitable’ version of theuniversality theorem for the Riemann zeta function, S. M. Voronin [ Vor3, Vor4 ]has established the functional independence of ζ ( s ), as we now explain: Theorem . Let z = ( z , z , ..., z n − ) ∈ C n and let N be the a nonnega-tive integer. If F ( z ) , F ( z ) ,..., F N ( z ) are continuous functions, not all vanishingsimultaneously, then there exists some s ∈ C such that N X k =0 s k F k ( ζ ( s ) , ζ ′ ( s ) , ..., ζ ( n − ( s )) = 0 . (2.3.4)Note that Theorem 2.6 implies that the Riemann zeta function does not sat-isfy any algebraic differential equation. As a result, ζ ( s ) is hypertranscendental . Universality of ζ ( s ) and the Riemann hypothesis In [
Bag1, Bag2 ], Bagchi showed that there is a connection between the loca-tion of the critical zeros of the Riemann zeta function and the universality of thisfunction. The corresponding result can be stated as follows: Here and in the sequel, we use the standard notation for the complex derivative (and higherorder derivatives) of an analytic function. We would like to attract the reader’s attention to the fact that the original proof of thehypertranscendence of the Riemann zeta function was due to Stadigh. Later on, such a proof wasgiven in a more general setting by Ostrowski, using a different mathematical approach (see [ Os ]). Theorem 2.6 provides an alternative proof of the solution of one of Hilbert’s famous problemsproposed in 1900 at the International Congress of Mathematicians in Paris.
HAFEDH HERICHI AND MICHEL L. LAPIDUS
Theorem . Let K be a compact subset of the vertical strip { < x < }× R ,with connected complement. Then, for any ǫ > , we have lim inf T →∞ T vol (cid:0)(cid:8) τ ∈ [0 , T ] : sup s ∈ K | ζ ( s ) − ζ ( s + iτ ) | < ǫ (cid:9)(cid:1) > if and only if the Riemann hypothesis ( RH ) is true. Here, as before, vol denotesthe Lebesgue measure on R . As a result, the universality of ζ ( s ) can be used to reformulate RH. Remark . Note that if the Riemann hypothesis is true, then ζ ( s ) is a non-vanishing analytic function in the right critical strip { < Re ( s ) < } . Hence, thefact that Equation ( ) holds follows at once from Voronin’s extended theorem ( Theorem 2.2 ) applied to g ( s ) := ζ ( s ) . The converse direction can be established byreasoning by contradiction and applying Rouch´e’s theorem ( from complex analysis ) about the number of zeros of the perturbation of an analytic function. Approximation by Taylor polynomials of ζ ( s )Another interesting application of the universality theorem for the Riemannzeta function was given in the work of M. Gauthier and R. Clouatre [ GauCl ]. UsingVoronin’s universality theorem, these authors showed that every holomorphic func-tion on a compact subset K of the complex plane having a connected complement K c can be uniformly approximated by vertical translates of Taylor polynomials of ζ ( s ). Denote by H ( K ) the set of all complex-valued holomorphic functions in anopen neighborhood ω of K ⊂ C and by T fn, a ( z ) := P nk =0 f ( k ) k ! ( z − a ) k , the n -thTaylor polynomial of f centered at a , where z ∈ C . Then we have the followingresult: Theorem . Let K ⊂ C with K c connected. Let g ∈ H ( K ) and ǫ > . Then,for each z ∈ ω ∩ K c , there exists τ ∈ R and n ∈ N such that sup z ∈ K | g ( z ) − T ζn, z + iτ ( z + iτ ) | < ǫ. (2.3.6) Path integrals, quantum theory and Voronin’s universality theorem
A physical application of Voronin’s theorem about the universality of the Rie-mann zeta function was obtained by K. M. Bitar, N. N. Khuri and H. C. Ren. Withintheir framework, Voronin’s universality theorem was used to explore a new numer-ical approach (via suitable discrete discretizations) to path integrals in quantummechanics (see [
BitKuRen ]).We will see in § quantization ’ of the universality of the Riemann zetafunction was obtained in [ HerLa1 ]. It enables us to obtain an operator-theoreticextension of Theorem 2.2 on the universality of the Riemann zeta function. Thisquantization is obtained in terms of a ‘suitable’ truncated version a ( T ) c of thespectral operator a c (see § a ( T ) c ),a map that relates the geometry of fractal strings to their spectra. The studyof the spectral operator was suggested by M. L. Lapidus and M. van Frankenhui-jsen in their development of the theory of complex dimensions in fractal geometry Their result was suggested in 2006 by Walter Hayman as a possible step toward attemptingto prove the Riemann hypothesis (see [
GauCl ]).
RUNCATED INFINITESIMAL SHIFTS AND QUANTIZED UNIVERSALITY OF ζ ( s ) 9 [ La-vF1, La-vF2, La-vF3, La-vF4 ]. Later on, it was thoroughly investigated(within a rigorous functional analytic framework) in [
HerLa1 ] and surveyed in thepapers [
HerLa2, HerLa3 ]. In the next section, we start by introducing the class ofgeneralized fractal strings and then define the spectral operator for fractal strings.
3. Generalized Fractal Strings and the Spectral Operator a = ζ ( ∂ c )In this section, we first recall (in § La-vF1, La-vF2, La-vF3, La-vF4 ], for ex-ample, in order to obtain general explicit formulas applicable to various aspectsof number theory, fractal geometry, dynamical systems and spectral geometry. In § a (as given in [ La-vF3, La-vF4 ]), we rigorously define a = a c as well asthe infinitesimal shift ∂ = ∂ c as unbounded normal operators acting on a scaleof Hilbert spaces H c parametrized by a nonnegative real number c , as was donein [ HerLa1, HerLa2, HerLa3 ]. Finally in § A generalizedfractal string η is defined as a local positive or complex measure on (0 , + ∞ ) satisfy-ing | η | (0 , x ) = 0, for some x >
0. For instance, if we consider the ordinary fractalstring L = { l j } ∞ j =1 with integral multiplicities w j , then the measure η L associatedto L is a standard example of a generalized fractal string: η L := ∞ X j =1 w j δ { l − j } , (3.1.1)where δ { x } is the Dirac delta measure (or the unit point mass) concentrated at x > l j are the scales associated to the connected components (or open intervals)constituting the ordinary fractal string L . We refer the reader to [ La-vF1, La-vF2,La-vF3, La-vF4 ] for more information about the theory of ordinary fractal stringsand many of its applications.
Remark . In contrast to an ordinary fractal string, the multiplicities w j ofa generalized fractal string η may be non-integral. For example, the prime string η := X m ≥ , p (log p ) δ { p m } (3.1.2)( where p runs over all prime numbers ) , is clearly a generalized ( and not an ordinary ) fractal string. Furthermore, observe that the multiplicities need not be positive num-bers either. A simple example of this situation is the M¨obius string η µ := ∞ X j =1 µ ( j ) δ { j } , (3.1.3) Here, the positive (local) measure | η | is the total variation measure of η . (For a review ofstandard measure theory, see, e.g., [ Coh, Fo ].) An ordinary fractal string L is a bounded open subset of the real line. Such a set consistsof countably many (bounded) open intervals { ( a j , b j ) } ∞ j , the lengths of which are denoted by l j = b j − a j , for each j ≥
1. We simply write L = { l j } ∞ j =1 . For instance, the Cantor string,defined as the complement of the ternary Cantor set in the interval [0 , L CS = { l j } ∞ j =1 , where the corresponding lengthsare l j = 3 − j with the corresponding multiplicities w j = 2 j − , for each j ≥ where µ ( j ) is the M¨obius function. This string can be viewed as the measure asso-ciated to the fractal string L µ = { j − } ∞ j =1 with real multiplicities w j = µ ( j ) . Notethat this string is not an ordinary fractal string. As a result, the use of the word‘generalized’ is well justified for this class of strings. The dimension D η of a generalized fractal string η is the abscissa of conver-gence of the Dirichlet integral R ∞ x − s η ( dx ): D η := inf (cid:26) σ ∈ R : Z ∞ x − σ | η | ( dx ) < ∞ (cid:27) , (3.1.4)The geometric counting function of η is N η ( x ) := Z x η ( dx ) = 12 ( η (0 , x ) + η (0 , x ]) . (3.1.5)The geometric zeta function associated to η is the Mellin transform of η : ζ η ( s ) := Z ∞ x − s η ( dx ) for Re ( s ) > D η . (3.1.6)From now on, we will assume that ζ η has a meromorphic extension to asuitable open connected neighborhood W of the half-plane of absolute convergenceof ζ η ( s ) (i.e., { Re ( s ) > D η } ). The set D η ( W ) of visible complex dimensions of η isdefined by D η ( W ) := { ω ∈ W : ζ η has a pole at ω } . (3.1.7)For instance, the geometric zeta function of the M¨obius string (see Equation(3.1.3)) is given by ζ µ ( s ) = ∞ X j =1 µ ( j ) j s = 1 ζ ( s ) , (3.1.8)the reciprocal of the Riemann zeta function.The spectral measure ν associated to η is defined by ν ( A ) = ∞ X k =1 η (cid:0) Ak (cid:1) , (3.1.9)for any bounded Borel set A ⊂ (0 , + ∞ ). Then N ν is called the spectral countingfunction of η .The spectral zeta function ζ ν associated to η is the geometric zeta functionassociated to ν . It turn out (as shown in [ La-vF1, La-vF2, La-vF3, La-vF4 ])that the spectral zeta function and the geometric zeta function of a generalizedfractal string η are related via the following formula: ζ ν ( s ) = ζ η ( s ) .ζ ( s ) , (3.1.10) The M¨obius function µ ( j ) equals 1 if j is a square-free positive integer with an even numberof prime factors, -1 if j is a square-free positive integer with an odd number of prime factors, and0 if j is not square-free. In the special case of ordinary fractal strings, Equation (3.1.10) was first observed in[
La2 ] and used in [
La3, La4, LaPo1, LaPo2, LaMa1, LaMa2, La-vF1, La-vF2, La-vF3,La-vF4, Tep, La5, LalLa1, LalLa2, HerLa1, HerLa2, HerLa3 ]. Furthermore, a formulaanalogous to the one given in Equation (3.1.10) exists for other generalizations of ordinary fractalstrings, including the class of fractal sprays (also called higher-dimensional fractal strings); see[
LaPo3, La2, La3, La-vF1, La-vF2, La-vF3, La-vF4 ]. RUNCATED INFINITESIMAL SHIFTS AND QUANTIZED UNIVERSALITY OF ζ ( s ) 11 where the factor ζ ( s ) is the Riemann zeta function.Next, we introduce two generalized fractal strings which will play an impor-tant role in the definition of the spectral operator, its operator-valued prime factorsand operator-valued Euler product, namely, the harmonic generalized fractal string h := ∞ X k =1 δ { k } , (3.1.11)and for a fixed but arbitrary prime p ∈ P , the prime harmonic string h p := ∞ X k =1 δ { p k } . (3.1.12)These two generalized fractal strings are related via the multiplicative convolutionof measures ∗ as follows (henceforth, P denotes the set of all prime numbers): h = ∗ p ∈P h p . (3.1.13)As a result, we have ζ h ( s ) = ζ ∗ h p p ∈P ( s ) = ζ ( s ) = Y p ∈P − p − s = Y p ∈P ζ h p ( s ) , (3.1.14)for Re ( s ) > η , its spectral measure ν is related to h and h p via the following formula (where ∗ denotes the multiplicative convolutionon (0 , + ∞ )): ν = η ∗ h = η ∗ (cid:0) ∗ p ∈P h p (cid:1) = ∗ p ∈P ν p , (3.1.15)where ν p = η ∗ h p is the spectral measure of h p for each p ∈ P . Note that by apply-ing the Mellin transform to the first equality of this identity, one recovers Equation(3.1.10).In their development of the theory of fractal strings in fractal geometry, thesecond author and M. van Frankenhuijsen obtained explicit distributional formulas associated to a given generalized fractal string η . These explicit formulas express thek-th distributional primitive (or anti-derivative) of η (when viewed as a distribution)in terms of its complex dimensions. For simplicity and given the needs of our func-tional analytic framework, we will only present these formulas in a restricted settingand for the case of a strongly languid generalized fractal string. We encourage thecurious reader to look at [
La-vF4 , § § Note that the original explicit formula was first obtained by Riemann in [
Rie ] as ananalytical tool aimed at understanding the distribution of the primes. We refer the reader to[
Edw, Ing, Ivi, Pat, Tit ] for more details about Riemann’s explicit formula and also to Appen-dix A in [
HerLa2 ] in which we provide a discussion of Riemann’s explicit original formula andthe explicit distributional formula obtained in [
La-vF4 , § § Roughly speaking, a generalized fractal string η is said to be strongly languid if its geometriczeta function ζ η satisfies some suitable polynomial growth conditions; see [ La-vF4 , § Given a strongly languid generalized fractal string η , and applying the explicitdistributional formula at level k = 0, we obtain an explicit representation of η , calledthe density of geometric states formula (see [ La-vF4 , § η = X ω ∈D η ( W ) res ( ζ η ( s ); ω ) x ω − . (3.1.16)Applying the explicit distributional formula (at the same level k = 0) to the spectralmeasure ν = η ∗ h , we obtain the following representation of ν , called the density ofspectral states formula (or density of frequencies formula ) (see [ La-vF4 , § ν = ζ η (1) + X ω ∈D η ( W ) res ( ζ η ( s ); ω ) ζ ( ω ) x ω − . (3.1.17) Remark . Many applications and extensions of fractal string theory and/orof the corresponding theory of complex fractal dimensions can be found through-out the books [ La-vF2, La-vF3, La-vF4, La-vF5, La5 ] and in [ La1, La2, La3,La4, LaPo1, LaPo2, LaPo3, LaMa1, LaMa2, HeLa, La-vF1, HamLa, Tep,LaPe, LaPeWi, LaLeRo, ElLaMaRo, LaLu1, LaLu2, LaLu-vF1, LaLu-vF2,LalLa1, LalLa2, LaRaZu, HerLa1, HerLa2, HerLa3, HerLa4, La6 ] . Theseinclude, in particular, applications to various aspects of number theory and arith-metic geometry, dynamical systems, spectral geometry, geometric measure theory,noncommutative geometry, mathematical physics and nonarchimedean analysis. ∂ c of the realline. The spectral operator was introduced by the second author and M. vanFrankenhuijsen in [
La-vF3 , § heuristically ’ as the map that sends the geometry ontothe spectrum of generalized fractal strings, but without providing the proper func-tional analytic framework needed to study it. (See [ La-vF3 , § La-vF4 , § HerLa1 ]. It was also surveyed in the papers[
HerLa2, HerLa3 ]. We will start by first defining the spectral operator and itsoperator-valued Euler product.Given a generalized fractal string η , and in light of the distributional explicitformulas [ La-vF4 , Theorems 5.18 & 5.22] (see Equations (3.1.16) and (3.1.17)above), the spectral operator a was heuristically defined as the operator mappingthe density of geometric states of η to its density of spectral states : η ν. (3.2.1)Now, considering the level k = 1 (in the explicit distributional formula), thespectral operator a = a c will be defined on a suitable weighted Hilbert space H c as For simplicity, we assume here that all of the complex dimensions are simple poles of ζ η and are different from 1. That is, roughly speaking, take “the antiderivative” of the corresponding expressions. The Hilbert space H c will depend on a parameter c ∈ R which appears in the weight µ c ( t ) = e − ct dt defining H c (see Equation (3.2.10) and the text surrounding it). As a result, thespectral operator will be denoted in the rest of the paper by either a or a c . RUNCATED INFINITESIMAL SHIFTS AND QUANTIZED UNIVERSALITY OF ζ ( s ) 13 the operator mapping the geometric counting function N η onto the spectral countingfunction N ν : N η ( x ) ν ( N η )( x ) := N ν ( x ) = ∞ X n =1 N η (cid:16) xn (cid:17) . (3.2.2)Using the change of variable x = e t , where t ∈ R and x >
0, we obtain an additive representation of the spectral operator a , f ( t ) a ( f )( t ) = ∞ X n =1 f ( t − log n ) . (3.2.3)For each prime p ∈ P , the operator-valued Euler factors a p are given by f ( t ) a p ( f )( t ) = ∞ X m =0 f ( t − m log p ) . (3.2.4)All of these operators are related by an Euler product as follows: f ( t ) a ( f )( t ) = Y p ∈P a p ( f )( t ) , (3.2.5)where the product is the composition of operators.Using the Taylor series representation of the function f , f ( t + h ) = e h ddt ( f )( t ) = e h∂ ( f )( t ) , (3.2.6)where ∂ = ddt is the infinitesimal shift of the real line (or also the first order differ-ential operator), we obtain the following representations of the spectral operator: a ( f )( t ) = ∞ X n =1 e − (log n ) ∂ ( f )( t ) = ∞ X n =1 (cid:18) n ∂ (cid:19) ( f )( t )= ζ ( ∂ )( f )( t ) = ζ h ( ∂ )( f )( t ) . (3.2.7)For all primes p , the operator-valued prime factors are given by a p ( f )( t ) = ∞ X m =0 f ( t − m log p ) = ∞ X m =0 e − m (log p ) ∂ ( f )( t ) = ∞ X m =0 (cid:0) p − ∂ (cid:1) m ( f )( t )= (cid:18) − p − ∂ (cid:19) ( f )( t ) = (1 − p − ∂ ) − ( f )( t ) = ζ h p ( ∂ )( t ) , (3.2.8)and hence, the operator-valued Euler product of the spectral operator a is given by a = Y p ∈P (1 − p − ∂ ) − ( f )( t ) . (3.2.9) Here, we will assume for pedagogical reasons that the complex-valued function f is infinitelydifferentiable. Of course, this is not always the case for an arbitrary generalized fractal string (sincethe atoms of η create discontinuities in the geometric counting function f := N η , for instance). Thisissue is addressed in [ HerLa1 ] by carefully studying the associated semigroup of operators; seeProposition 3.20 below.
Remark . Within our functional analytic framework, and in light of theabove new representation of the spectral operator a c , its operator-valued prime fac-tors and operator-valued Euler product, the function f will not necessarily representthe ( geometric or spectral ) counting function of some generalized fractal string η butwill instead be viewed as an element of the weighted Hilbert space H c which was in-troduced in [ HerLa1 ] in order to study the spectral operator ( following, but alsosuitably modifying, a suggestion originally made in [ La-vF3 ]) . Remark . The above representation of the spectral operator given in Equa-tion ( ) was justified in the functional analytic framework provided in [ HerLa1 ] .Furthermore, the representation of the operator-valued Euler product Q p ∈P a p ( seeEquation ( )) and its operator-valued prime factors a p ( see Equation ( )) were justified in [ HerLa4 ] . In particular, for c > , the Euler product convergesin the operator norm ( of the Hilbert space H c ) . We note that the operator-valuedEuler product associated to a c ( see Equation ( )) is conjectured to converge ( in an appropriate sense ) to a c in the critical strip; that is, for < c < . ( See [ La-vF3 , § ] or [ La-vF4 , § ] . ) In [ HerLa4 ] , a ‘ suitable mode of conver-gence ’ will be considered in order to address this conjecture. A full discussion of theoperator-valued prime factors and operator-valued Euler product will be omitted inthis paper. Instead, we will focus on studying some of the properties of the spectraloperator a c and of its truncations a ( T ) c , as well as of the infinitesimal shift ∂ c andits truncations ∂ ( T ) c , which are the central characters of the present paper. Indeed,they will play a key role in our quantization of the universality of the Riemannzeta function, as will be discussed in § ( See § )The spectral operator a c was rigorously defined in [ HerLa1 ] as an unboundedlinear operator acting on the weighted Hilbert space H c = L ( R , µ c ( dt )) , (3.2.10)where c ≥ µ c is the absolutely continuous measure on R given by µ c ( dt ) := e − ct dt , with dt being the standard Lebesgue measure on R . More precisely, a c isdefined by a c ( f )( t ) = ζ ( ∂ c )( f )( t ) . (3.2.11)In view of Equation (3.2.11), the infinitesimal shift ∂ c clearly plays an importantrole in our proposed definition of the spectral operator. The domain of ∂ c is givenby D ( ∂ c ) = { f ∈ H c ∩ AC ( R ) : f ′ ∈ H c } , (3.2.12)where AC ( R ) is the space of (locally) absolutely continuous functions on R and f ′ denotes the derivative of f , viewed either as an almost everywhere defined functionor as a distribution. Furthermore, the domain of the spectral operator is given by D ( a c ) = { f ∈ D ( ∂ c ) : a c ( f ) = ζ ( ∂ c )( f ) ∈ H c } . (3.2.13)Moreover, for f ∈ D ( ∂ c ), we have that ∂ c ( f )( t ) := f ′ ( t ) = dfdt ( t ) , (3.2.14) For information about absolutely continuous functions and their use in Sobolev theory, see,e.g., [ Fo ] and [ Br ]; for the theory of distributions, see, e.g., [ Schw ], [ Ru ] and [ Br ]. RUNCATED INFINITESIMAL SHIFTS AND QUANTIZED UNIVERSALITY OF ζ ( s ) 15 almost everywhere (with respect to dt , or equivalently, with respect to µ c ( dt )). Inshort, we write that ∂ c ( f ) = f ′ , where the equality holds in H c .Finally, Equation (3.2.11) holds for all f ∈ D ( a c ), and ζ ( ∂ c ) is interpretedin the sense of the functional calculus for unbounded normal operators (see, e.g.,[ Ru ]). Indeed, as is recalled in Theorem 3.14 below, we show in [ HerLa1 ] that theoperator a c is normal (i.e., that it is a closed, possibly unbounded operator, andthat it commutes with its adjoint). (It is bounded for c >
1, but unbounded for0 < c ≤
1, as is also shown in [
HerLa1, HerLa2, HerLa3 ].)
Remark . The weighted Hilbert space H c is the space of ( C -valued ) Lebesguesquare-integrable functions f ( on R ) with respect to the weight function w ( t ) = e − ct ; namely, as was stated in Equation ( ) , H c = L ( R , e − ct dt ) . It is equippedwith the inner product < f, g > c := Z R f ( t ) g ( t ) e − ct dt, (3.2.15) where g denotes the complex conjugate of g . Therefore, the norm of an element of H c is given by || f || c := (cid:18) Z R | f ( t ) | e − ct dt (cid:19) . (3.2.16)It is shown in [ HerLa1 ] that the functions which lie in the domain of ∂ c (and hence also those belonging to the domain of a c , ∂ ( T ) c or a ( T ) c ) satisfy ‘ naturalboundary conditions ’. Namely, if f ∈ D ( ∂ c ), then | f ( t ) | e − ct → t → ±∞ . (3.2.17) Remark . The boundary conditions given in Equation ( ) are satisfiedby any function f in D ( ∂ c ) ( see Equation ( )) or in the domain of a functionof ∂ c such as the spectral operator a c = ζ ( ∂ c ) ( see Equation ( )) , as wellas the truncated infinitesimal shifts ∂ ( T ) c ( see Equation ( )) and the truncatedspectral operators a ( T ) c ( see Equation ( )) . Furthermore, clearly, if η representsan ordinary fractal string L , then N L vanishes identically to the left of zero, and if,in addition, the ( Minkowski ) dimension of L is strictly less than c , then it followsfrom the results of [ LaPo2 ] that N η ( t ) = o ( e ct ) as t → + ∞ ( i.e., N η ( x ) = o ( x c ) as x → + ∞ , in the original variable x = e t ) , so that f := N η then satisfies theboundary conditions given by ( ) . Remark . Note that it also follows from the results of [ LaPo2 ] ( see also [ La-vF4 ]) that for c in the ‘critical interval’ (0 , , the parameter c can be inter-preted as the least upper bound for the ( Minkowski ) fractal dimensions of the allowedunderlying fractal strings. An intrinsic connection between the representations obtained in Equation(3.2.7) was given in [
HerLa1 ]. Our next result justifies the representation of the For the notion of Minkowski dimension, see, e.g., [
Man, La1, Fa, Mat, La-vF4 ]. Recallthat it was observed in [
La2 ] (using a result of [
BesTa ]) that for an ordinary fractal string L (represented by a bounded open set Ω ⊂ R ), the abscissa of convergence of ζ L coincides with theMinkowski dimension of L (i.e., of ∂ Ω); for a direct proof of this result, see [
La-vF4 , Theorem1.10]. spectral operator a c as the composition map of the Riemann zeta function and theinfinitesimal shift ∂ c (see Equation (3.2.7)): Theorem . Assume that c > . Then, a can be uniquely extended to abounded operator on H c and, for any f ∈ H c , we have a ( f )( t ) = ∞ X n =1 f ( t − log n ) = ζ ( ∂ c )( f )( t ) = ∞ X n =1 n − ∂ c ! ( f )( t ) , (3.2.18) where the equalities hold for almost all t ∈ R as well as in H c . Remark . In other words,
Theorem 3.8 justifies the ‘heuristic’ representa-tion of the spectral operator given above in Equation ( ) . Indeed, it states thatfor c > , we have a c = ζ ( ∂ c ) = ∞ X n =1 n − ∂ c , (3.2.19) where the equality holds in B ( H c ) , the space of bounded linear operators on H c . Remark . In addition, it is shown in [ HerLa1 ] that for any c > , andfor all f in a suitable dense subspace of D ( a c ) ( and hence, of H c ) , an appropriate‘analytic continuation’ of Equation ( ) continues to hold ( when applied to f ) . A detailed study of the invertibility (and also, of the quasi-invertibility )of the spectral operator a c is provided in [ HerLa1 ]. It was also surveyed in thepapers [
HerLa2, HerLa3 ]. In particular, in that study, using the functional cal-culus along with the spectral mapping theorem for unbounded normal operators(the continuous version when c = 1 and the meromorphic version, when c = 1),a precise description of the spectrum σ ( a c ) of the spectral operator is obtained in[ HerLa1 ]. More explicitly, we show that σ ( a c ) is equal to the closure of the rangeof the Riemann zeta function on the vertical line L c = { λ ∈ C : Re ( λ ) = c } : Theorem . [ HerLa1 ] Assume that c ≥ . Then σ ( a ) = ζ ( σ ( ∂ )) = cl (cid:0) ζ ( { λ ∈ C : Re ( λ ) = c } ) (cid:1) , (3.2.20) where σ ( a ) is the spectrum of a = a c and N = cl ( N ) is the closure of N ⊂ C . In [
HerLa1 ] (see also [
HerLa3 ]), a spectral reformulation of the Riemannhypothesis is obtained, further extending from an operator-theoretic point of viewthe earlier reformulation of RH obtained by the second author and H. Maier in theirstudy of the inverse spectral problem for fractal strings (see [
LaMa1, LaMa2 ]),in relation to answering the question (`a la Mark Kac [
Kac ], but interpreted in avery different sense)“Can one hear the shape of a fractal string?”.
Theorem . [ HerLa1 ] The spectral operator a = a c is quasi-invertible forall c ∈ (0 , − ( or equivalently, for all c ∈ ( , if and only if the Riemannhypothesis is true. The spectral operator a c is said to be quasi-invertible if its truncation a ( T ) is invertible forevery T >
0. See [
HerLa1, HerLa2 ] for a more detailed discussion of the quasi-invertibility of a c , along with § a ( T ) , where T ≥ RUNCATED INFINITESIMAL SHIFTS AND QUANTIZED UNIVERSALITY OF ζ ( s ) 17 Remark . The above theorem also enables us to give a proper formulationof [ La-vF4 , Corollary 9.6 ] in terms of the quasi-invertibility of the spectral operator a c , as interpreted in [ HerLa1, HerLa2, HerLa3 ] . In the next subsection, we will discuss some of the fundamental properties(obtained in [
HerLa1, HerLa2 ]) of the infinitesimal shift ∂ = ∂ c and the associatedstrongly continuous group { e − t∂ } t ≥ . ∂ c . The infinitesimal shift ∂ = ∂ c , with c ≥ ∂ ( T ) = ∂ ( T ) c , T ≥
0, to be discussed in § HerLa1, HerLa2,HerLa3, HerLa4 ] as well as in the present paper, as we shall see, in particular,in § Theorem . [ HerLa1 ] The infinitesimal shift ∂ = ∂ c is an unboundednormal linear operator on H c . Moreover, its adjoint A ∗ is given by ∂ ∗ = 2 c − ∂, with D ( ∂ ∗ ) = D ( ∂ ) . (3.3.1) Remark . The residual spectrum, σ r ( A ) , of a normal ( possibly unboun-ded ) linear operator A : D ( A ) ⊂ H → H , where D ( A ) is the domain of A and H issome complex Hilbert space, is empty. Hence, the essential spectrum of A , σ e ( A ) ,which consists of all the approximate eigenvalues of A , is equal to the entirespectrum of A , denoted by σ ( A ) . Remark . References on the spectral theory ( and the associated functionalcalculus ) of unbounded linear operators, with various degrees of generality and em-phasis on the applications of the theory, include [ DunSch, Kat, Ru, ReSi, Sc,JoLa, Ha ] . In particular, the case of unbounded normal operators ( which is of mostdirect interest here ) is treated in Rudin’s book [ Ru ] . Theorem . [ HerLa1 ] The spectrum, σ ( ∂ ) , of the differentiation operator ( or infinitesimal shift ) ∂ = ∂ c is equal to the closed vertical line of the complexplane passing through c ≥ ; furthermore, it coincides with the essential spectrumof ∂ : σ ( ∂ ) = σ e ( ∂ ) = { λ ∈ C : Re ( λ ) = c } , (3.3.2) where σ e ( ∂ ) consists of all the approximate eigenvalues of ∂ . Moreover, the pointspectrum of ∂ is empty ( i.e., ∂ does not have any eigenvalues ) , so that the spectrumof ∂ is purely continuous. Corollary . For any c ≥ , we have σ ( ∂ ∗ ) = σ ( ∂ ) = c + i R . In light of Corollary 3.18, the following result is really a consequence of The-orem 3.14 and will be very useful to us in the sequel (see § § Corollary . For any c ≥ , we can write ∂ = c + iV and ∂ ∗ = c − iV, (3.3.3) where c = cI = Re ( ∂ ) ( a constant multiple of the identity operator on D ( ∂ ) ⊂ H c ) and V = Im ( ∂ ) , an unbounded self-adjoint operator on H c , with domain D ( V ) = D ( ∂ ) = D ( ∂ ∗ ) . Moreover, the spectrum of V is given by σ ( V ) = R . Recall that λ ∈ C is called an approximate eigenvalue of A if there exists a sequence ofunit vectors { ψ n } ∞ n =1 of H such that ( A − λ ) ψ n → n → ∞ . (See, e.g., [ Sc ].) The next result (also from [
HerLa1 ]) enables us to justify the use of the term“infinitesimal shift”(when referring to the operator ∂ = ∂ c ) as well as some of theformal manipulations occurring in Equations (3.2.6), (3.2.7), (3.2.8) and (3.2.9) of § Proposition . Fix c ≥ and write ∂ = ∂ c . Then, the following two prop-erties hold :( i ) { e − t∂ } t ≥ is a strongly continuous contraction semigroup of bounded linearoperators on H c and || e − t∂ || = e − tc for any t ≥ . Hence, its infinitesimal generator ∂ is an m -accretive operator on H c ( in the sense of [ Kat, JoLa, Paz ]) . ( ii ) { e − t∂ } t ≥ is a translation ( or shift ) semigroup. That is, for every t ≥ , ( e − t∂ )( f )( u ) = f ( u − t ) , for all f ∈ H c and u ∈ R . ( For a fixed t ≥ , this equalityholds between elements of H c and hence, for a.e. u ∈ R . ) Remark . An entirely analogous result holds for the semigroup { e t∂ } t ≥ ,except that it is then an expanding ( rather than a contraction ) semigroup. Similarly,for any c ∈ R such that c ≤ , all of the results stated in §
4. The Truncated Spectral Operators a ( T ) c = ζ ( ∂ ( T ) c )As was alluded to above (at the beginning of § § § § ∂ ( T ) c and their properties. Re-call from Corollary 3.19 that for any c ≥
0, the infinitesimal shift ∂ = ∂ c is givenby ∂ = c + iV, where V := Im ( ∂ ) (the imaginary part of ∂ ) is an unbounded self-adjoint operatorsuch that σ ( V ) = R .Given T ≥
0, we define the T - truncated infinitesimal shift as follows: ∂ ( T ) := c + iV ( T ) , (4.1.1)where V ( T ) := φ ( T ) ( V ) (4.1.2)(in the sense of the functional calculus), and φ ( T ) is a suitable (i.e., T -admissible)continuous (if c = 1) or meromorphic (if c = 1) cut-off function chosen so that φ ( T ) ( R ) = c + i [ − T, T ]).The next result states that the spectrum σ ( ∂ ( T ) c ) of the truncated infinitesi-mal shift is equal to the vertical line segment of height T and abscissa c , symetrically Detailed information about the theory of semigroups of bounded linear operators can befound, e.g., in the books [
HiPh, EnNa, Paz, Kat, ReSi, JoLa ]. RUNCATED INFINITESIMAL SHIFTS AND QUANTIZED UNIVERSALITY OF ζ ( s ) 19 located with respect to c . Theorem . [ HerLa1 ] For any
T > and c ≥ , the spectrum σ ( ∂ ( T ) ) ofthe truncated infinitesimal shift ∂ ( T ) = ∂ ( T ) c is given by σ ( ∂ ( T ) ) = { c + iτ : | τ | ≤ T, τ ∈ R } = c + i [ − T, T ] . (4.1.3) Moreover, the spectrum σ ( V ( T ) ) of the imaginary part V ( T ) of the infinitesi-mal shift is given by σ ( V ( T ) ) = [ − T, T ] . (4.1.4) Next, wedefine our other main objects of study, the truncated spectral operators (denotedby a ( T ) ) which, along with the truncated infinitesimal shifts ∂ ( T ) c introduced in § c ≥
0. Then, given T ≥
0, the T - truncated spectral operator is defined asfollows: a ( T ) := ζ (cid:16) ∂ ( T ) (cid:17) . (4.2.1)More precisely, in the definition of ∂ ( T ) = c + iV ( T ) , with V ( T ) = φ ( T ) ( V ),as given in Equation (4.1.1) and Equation (4.1.2), the T -admissible function φ ( T ) is chosen as follows:(i) If c = 1, φ ( T ) is any continuous function such that φ ( T ) ( R ) = [ − T, T ]. (Forexample, φ ( T ) ( τ ) = τ for 0 ≤ τ ≤ T and φ ( T ) ( τ ) = T for τ ≥ T ; also, φ ( T ) is odd.)(ii) If c = 1 (which corresponds to the pole of ζ ( s ) at s = 1), then φ ( T ) isa suitable meromorphic analog of (i). (For example, φ ( T ) ( s ) = Tπ tan − ( s ), so that φ ( T ) ( R ) = [ − T, T ].)One then uses the measurable functional calculus and an appropriate (con-tinuous or meromorphic, for c = 1 or c = 1, respectively) version of the spec-tral mapping theorem ( SMT ) for unbounded normal operators (as provided in[
HerLa1 , Appendix E]) in order to define both ∂ ( T ) and a ( T ) = ζ ( ∂ ( T ) ), as well asto determine their spectra (see § ∂ ( T ) ): SMT : σ ( ψ ( L )) = ψ ( σ ( L )) if ψ is a continuous ( resp., meromorphic ) function on σ ( L ) ( resp., on a connectedopen neighborhood of σ ( L )) and L is an unbounded normal operator. Remark . More precisely, in the meromorphic case, in the above equal-ity ( in the statement of SMT ) , one should exclude the poles of ψ which belongto σ ( L ) . Alternatively, one can view the meromorphic function ψ as a continuousfunction with values in the Riemann sphere e C := C ∪ {∞} and then write SMT inthe following simpler form : σ ( ψ ( L )) = ψ ( σ ( L )) . ( See [ HerLa1 , Appendix E] , along with the relevant references therein, including [ Ha ] . )Note that for c = 1 (resp., c = 1), ∂ ( T ) and a ( T ) are then continuous (resp.,meromorphic) functions of the normal (and sectorial, see [ Ha ]) operator ∂ . Anentirely analogous statement is true for the spectral operator a = ζ ( ∂ ). Theorem . [ HerLa1 ] ( i ) Assume that c ≥ , with c = 1 . Then, for all T ≥ , a ( T ) is a bounded normal linear operator. Furthermore, its spectrum σ ( a ( T ) ) is given by the following compact ( and hence, bounded ) subset of C : σ ( a ( T ) ) = { ζ ( c + iτ ) : | τ | ≤ T, τ ∈ R , τ = 0 } . (4.2.2)( ii ) When c = 1 , a similar statement holds for all T > except that now, a ( T ) is an unbounded ( i.e., not bounded ) normal operator with spectrum given by ( with cl denoting the closure of a set ) σ ( a ( T ) ) = cl { ζ (1 + iτ ) : | τ | ≤ T, τ ∈ R } , (4.2.3) a non-compact ( and in fact, unbounded ) subset of C . Alternately, one could write e σ ( a ( T ) ) = { ζ (1 + iτ ) : | τ | ≤ T, τ ∈ R } , (4.2.4) a compact subset of the Riemann sphere e C = C ∪ {∞} , where ζ is viewed as a ( continous ) e C -valued function and the extended spectrum e σ ( a ( T ) ) of a ( T ) is givenby e σ ( a ( T ) ) := σ ( a ( T ) ) ∪ {∞} ( still when c = 1) . Proof.
We include this particular proof in order to illustrate the use of thespectral mapping theorem
SMT discussed in Remark 4.2 and the text preceding it.In light of Equation (4.2.1), this follows immediately from the continuous ( c =1) or meromorphic ( c = 1) version of the spectral mapping theorem, according towhich σ ( a ( T ) ) = σ ( ζ ( ∂ ( T ) )) = cl { ζ ( σ ( ∂ ( T ) )) } . (4.2.5)Now, it follows from the results of § σ ( ∂ ( T ) ) = { c + iτ : | τ | ≤ T, τ ∈ R } . (4.2.6)Therefore, combining Equations (4.2.5) and (4.2.6), we obtain σ ( a ( T ) ) = cl ( ζ ( c + iτ : | τ | ≤ T, τ ∈ R } ) . (4.2.7)Note that if c = 1, then ζ is continuous on the vertical line Re ( s ) = c andhence, on the compact vertical segment given by Equation (4.2.6). Hence, its rangealong this segment is compact and therefore closed in C . This explains why we donot need to include the closure in Equation (4.2.2) giving the expression of the spec-trum σ ( a ( T ) ) of a ( T ) when c = 1. Moreover, since σ ( a ( T ) ) is compact and hence,bounded, the truncated infinitesimal shift a ( T ) is a bounded operator. When c = 1 (i.e., in case (i)), we have that e σ ( a ( T ) ) = σ ( a ( T ) ) since a ( T ) is then bounded;see, e.g., [ Ha ] for the notion of extended spectrum. In short, e σ ( L ) := σ ( L ) if the linear operator L is bounded, and e σ ( L ) := σ ( L ) ∪ {∞} if L is unbounded; so that e σ ( L ) is always a compact subsetof e C . This theorem is applied to the function ζ and the bounded normal operator ∂ ( T ) studiedin § RUNCATED INFINITESIMAL SHIFTS AND QUANTIZED UNIVERSALITY OF ζ ( s ) 21 On the other hand, when c = 1, ζ is meromorphic in an open neighborhoodof the vertical line { Re ( s ) = c = 1 } , and has a (simple) pole at s = 1. It followsthat its range is unbounded along the vertical segment σ ( ∂ ( T ) ) = { iτ : | τ | ≤ T, τ ∈ R , τ = 0 } . (4.2.8)Therefore, still when c = 1, we must keep the closure in the expression for σ ( a ( T ) )given by Equation (4.2.7). In addition, as stated in part (ii) of the theorem, a ( T ) is an unbounded normal operator when c = 1 because its spectrum is unbounded(and thus, non-compact). (cid:3)
5. Truncated Infinitesimal Shifts and Quantum Universality of ζ ( s )In the present section, we provide a ‘quantum’ analog of the universality ofthe Riemann zeta function ζ = ζ ( s ). Somewhat surprisingly at first, in our context,the proper formulation of Voronin’s universality theorem (and its various gener-alizations) does not simply consists in replacing the complex variable s with theinfinitesimal shift ∂ = ∂ c . In fact, as we shall see, one must replace the complexvariable s with the family of truncated infinitesimal shifts ∂ ( T ) = ∂ ( T ) c (with T ≥ c ≥ a = ζ ( ∂ ) is universal (and since ζ is a highly nonlinear function, wecould not say either that the family of truncated spectral operators a ( T ) = ζ ( ∂ ( T ) )is universal). Instead, the proper statement of ‘quantum universality’ is directlyexpressed in terms of the truncated infinitesimal shifts ∂ ( T ) and their imaginarytranslates. The “uni-versality”of the spectral operator a = ζ ( ∂ ) roughly means that any non-vanishingholomorphic function of ∂ on a suitable compact subset of the right critical strip { < Re ( s ) < } can be approximated (in the operator norm) arbitrarily closely byimaginary translates of ζ ( ∂ ). More accurately, any such non-vanishing function ofthe truncated infinitesimal shifts ∂ ( T ) = ∂ ( T ) c can be uniformly (in the parameters c and T ) approximated (in the operator norm on H c ) by the composition of ζ andsuitable imaginary translates of ∂ ( T ) = ∂ ( T ) c .Indeed, we have the following operator-theoretic generalization of the extendedVoronin universality theorem , expressed in terms of the imaginary translates of the T - truncated infinitesimal shifts ∂ ( T ) = ∂ ( T ) c (with parameter c ).We begin by providing an operator-theoretic generalization of the universalitytheorem which is in the spirit of Voronin’s original universality theorem (Theorem2.1) and its extension (Theorem 2.2). Note that since the spectrum is always a closed subset of C , it is non-compact if and onlyif the operator is unbounded. On the other hand, when c = 1, the extended spectrum of a ( T ) isdefined by e σ ( a ( T ) ) = σ ( a ( T ) ) ∪ {∞} (since the operator a ( T ) is unbounded, see, e.g., [ Ha ]) and isa closed (and hence, compact) subset of the Riemann sphere e C . It is therefore still given by theright-hand side of Equation (4.2.4), but with ζ viewed as a continuous function with values in e C ,as is explained in Remark 4.2. Theorem . [ HerLa1 ] (Quantized universality of ζ ( s ); first version) . LetK be a compact subset of the right critical strip { < Re ( s ) < } of the followingform. Assume, for simplicity, that K = K × [ − T , T ] , for some T ≥ , where K isa compact subset of the open interval ( , .Let g : K → C be a non-vanishing ( i.e., nowhere vanishing ) continuous functionthat is holomorphic in ˚ K , the interior of K ( which may be empty ) . Then, given any ǫ > , there exists τ ≥ depending only on ǫ ) such that H op ( τ ) := sup c ∈K ,
For τ >
0, let K = K × [ − T, T ]. Then, we consider the following twocases:(i) If T = 0, then ˚ K = ∅ (interior in C ). Also, if ˚ K = ∅ (interior in R ), then˚ K = ∅ (interior in C ). In either case, we only need to know that g is continuouson K × [ − T , T ] ⊆ K × R . The remainder of the proof, however, proceeds exactlyas in part (ii) below, by applying Theorem 2.2 to the continuous function g andthe compact set K = K × [ − T , T ] with empty interior in C (as well as to therestriction of g to K T := K × [ − T, T ], with T such that 0 < T ≤ T ).(ii) If T > K 6 = ∅ , then ˚ K = ∅ . Then, we need to require that g isholomorphic in the interior of K = K × [ − T , T ], in addition to being continuouson K . Now, by the universality of the Riemann zeta function applied to the nowherevanishing function g : K → C (or, more specifically, by applying Theorem 2.2 to g : K → C , where K = K × [ − T , T ]), we conclude that given ǫ > τ = τ ( T , ǫ ) ≥ | g ( s ) − ζ ( s + iτ ) | ≤ ǫ, for all s ∈ K. (5.1.3)In addition, the set of such numbers τ has positive lower density.Next, given T such that 0 < T ≤ T , let us set K T = K × [ − T, T ], so that K = K T . Clearly, for any such T , we have K T ⊆ K T . Hence, since it follows fromTheorem 4.1 in § ∂ ( T ) is precisely determined), that for Note that the complement of K in C is connected (since the complement of K in R , beingan open subset of R , is an at most countable disjoint union of intervals). RUNCATED INFINITESIMAL SHIFTS AND QUANTIZED UNIVERSALITY OF ζ ( s ) 23 every c ∈ K , we have σ ( ∂ ( T ) ) = [ c − iT, c + iT ] = { c } × [ − T, T ] ⊆ σ ( ∂ ( T ) ) = [ c − iT , c + iT ] = { c } × [ − T , T ] ⊆ K = K T = K × [ − T , T ] . (5.1.4)We conclude that g is continuous (and thus certainly measurable) on σ ( ∂ ( T ) ).We can therefore apply to ∂ ( T ) = ∂ ( T ) c the continuous version of the functionalcalculus (for unbounded normal operators) to deduce that φ ( ∂ ( T ) c ) = g ( ∂ ( T ) c ) − ζ ( ∂ ( T ) c + iτ ) , (5.1.5)where φ ( s ) := g ( s ) − ζ ( s + iτ ), for s ∈ K ; so that, in light of Equation (5.1.3), | φ ( s ) | ≤ ǫ, for all s ∈ K. (5.1.6)(Recall that each operator ∂ ( T ) = ∂ ( T ) c is bounded on H c since it has a boundedspectrum.) This same functional calculus (or, equivalently, the corresponding ver-sion of the spectral theorem for possibly unbounded normal operators, see [ Ru ])implies that each of the operators g ( ∂ ( T ) c ), ζ ( ∂ ( T ) c + iτ ) and φ ( ∂ ( T ) c ) belongs to B ( H c ), and || φ ( ∂ ( T ) c ) || = sup s ∈ σ ( ∂ ( T ) c ) | φ ( s ) | ≤ sup s ∈ K | φ ( s ) | ≤ ǫ. (5.1.7)Note that the first inequality follows from the fact that σ ( ∂ ( T ) c ) ⊆ K while thesecond inequality follows from Equation (5.1.6). Since Equation (5.1.7) holds forevery c ∈ K and for every T such that 0 < T ≤ T , and recalling that φ ( ∂ ( T ) c ) isgiven by the identity (5.1.5) (which holds in B ( H c )), we conclude that given any ǫ > T ≥ τ = τ ( ǫ ) ≥ c ∈K , K ⊂ ( , 1) and that ∂ ( T ) = ∂ ( T ) c , so that ζ ( ∂ ( T ) c + iτ ) is a bounded (normal)operator. Similarly, φ is continuous on the compact set σ ( ∂ ( T ) c ) = [ c − iT, c + iT ], and hence φ ( ∂ ( T ) c ) is a bounded (normal) operator. In any case, for every c ∈ K , g is continuous and istherefore bounded on the compact set σ ( ∂ ( T ) c ), in agreement with Equation (5.1.6). Furthermore, g ( ∂ ( T ) c ) is a bounded (normal) operator and its norm satisfies an inequality implied by (5.1.7). We use here, in particular, the fact that ζ is continuous on the compact vertical segment σ ( ∂ ( T ) c + iτ ) = σ ( ∂ ( T ) c ) + iτ = [ c + i ( τ − T ) , c + i ( τ + T )] because c = 1 for c ∈ K . Definition . To say that K is vertically convex means that if c − iT ′ and c + iT belong to K for some c ∈ K and T ′ ≤ ≤ T , then the entire vertical linesegment [ c − iT ′ , c + iT ] is contained in K . Theorem . [ HerLa1 ] (Quantized universality of ζ ( s ); second, more general,version) . Let K be any compact, vertically convex subset of the right critical strip { < Re ( s ) < } , with connected complement in C . Assume, for simplicity, that K is symmetric with respect to the real axis. Denote by K the projection of K onto thereal axis, and for c ∈ K , let T ( c ) := sup ( { T ≥ c − iT, c + iT ] ⊂ K } ) , (5.2.1) and T ( c ) = −∞ if there is no such T . ( By construction, K is a compact subsetof ( , and ≤ T ( c ) < ∞ , for every c ∈ K . ) Further assume that c T ( c ) iscontinuous on K .Let g : K → C be a non-vanishing ( i.e., nowhere vanishing ) continuous functionthat is holomorphic in the interior of K ( which may be empty ) . Then, given any ǫ > , there exists τ ≥ depending only on ǫ ) such that J op ( τ ) := sup c ∈K , ≤ T ≤ T ( c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ( ∂ ( T ) c ) − ζ ( ∂ ( T ) c + iτ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ, (5.2.2) where ∂ ( T ) = ∂ ( T ) c is the T -truncated infinitesimal shift ( with parameter c ) and || . || denotes the usual norm in B ( H c ) ( the space of bounded linear operators on H c ) .In fact, the set of such τ ′ s has a positive lower density and, in particular,is infinite. More precisely , we have lim inf ρ → + ∞ ρ vol ( { τ ∈ [0 , ρ ] : J op ( τ ) ≤ ǫ } ) > . (5.2.3) Proof. Let N := { c + iT : T ∈ R , | T | ≤ T ( c ) , c ∈ K} . Assume that T T ( c )is continuous on K . Then, we claim that N is a compact subset of C (and in fact,of { s ∈ C : < Re ( s ) < } ).In order to justify this claim, we proceed as follows. Since N is clearly bounded,then it suffices to show that N is closed. Let ( c n , T n ) = c n + iT n be an infinite se-quence of elements of N such that( c n , T n ) → ( c, T ) = c + iT. Thus, c n ∈ K , c n → c and T n → T. As a result, c ∈ K (since K is compact, and hence is closed in R ). Also, since T n → T as n → ∞ and | T n | ≤ T ( c n ) , for all n ≥ , we have lim n →∞ | T n | = | T | ≤ lim sup n →∞ T ( c n ) . But since c n → c and the map u T ( u ) is continuous on K , we have that T ( c n ) → T ( c ) as n → ∞ . Hence, | T | ≤ T ( c ) for any c ∈ K and so ( c, T ) = c + iT ∈ N .The remainder of the proof of Theorem 5.4 proceeds much as in the proof ofTheorem 5.1, by applying the extended Voronin theorem (Theorem 2.2), combined RUNCATED INFINITESIMAL SHIFTS AND QUANTIZED UNIVERSALITY OF ζ ( s ) 25 with the functional calculus for bounded normal operators, and using the fact that σ ( ∂ ( T ) c ) = [ c − iT, c + iT ] is contained in σ ( ∂ ( T ) c ) = [ c − iT , c + iT ] for any T suchthat 0 < T ≤ T and any c ∈ K . (cid:3) Remark . Instead of assuming that K is symmetric with respect to the realaxis, it would suffice to suppose that c + iT ∈ K ( for some c ∈ K and T > impliesthat c − iT ∈ K , and vice versa. Remark . As in the scalar case ( and taking K to be a line segment ) , wesee that any continuous curve (cid:0) of ∂ ( T ) c (cid:1) can be approximated by imaginary trans-lates of a ( T ) = ζ ( ∂ ( T ) c ) . Hence, roughly speaking, the spectral operator a (cid:0) or its T -truncations a ( T ) (cid:1) can emulate any type of complex behavior: it is chaotic . Note that conditionally (i.e., under the Riemann hypothesis), and applyingthe above operator-theoretic version of the universality theorem to g ( s ) := ζ ( s ), wesee that, roughly speaking, arbitrarily small scaled copies of the spectral operatorare encoded within a itself. In other words, a (or its T -truncation) is both chaoticand fractal. 6. Concluding Comments The universality of the Riemann zeta function ζ ( s ) in the right critical strip { < Re ( s ) < } and its consequences play an important role in other parts ofour work in [ HerLa1 ] (see also [ HerLa2, HerLa3 ]). In particular, the density of ζ ( s ) along the vertical lines Re ( s ) = c (with < c < HerLa1 ]), implies that for < c < 1, the spectrum of thespectral operator a c = ζ ( ∂ c ) is equal to the whole complex plane: σ ( a c ) = C . Bycontrast, σ ( a c ) is a compact subset of C for c > GarSt ]) that σ ( a c ) is an unbounded, strict subset of C for 0 < c < . The latter result is a consequence of the “non-universality”of ζ ( s )on the left critical strip { < Re ( s ) < } ; see [ GarSt ] and the relevant referencestherein.Our study of the truncated infinitesimal shifts ∂ ( T ) c and their spectra (seeTheorem 4.1) has played a crucial role in our proposed quantization of the uni-versality of the Riemann zeta function obtained in Theorems 5.1 and 5.4. Notethat, in light of our functional analytic framework, one should be able to obtainin a similar manner further operator-theoretic versions (or ‘quantizations’) of theknown extensions of Voronin’s theorem about the universality of ζ ( s ) and other L -functions (see, for example, Appendix B, Theorems 8.3, 8.6 and 8.7 below). Inthis broader setting, we expect that the complex variable s should still be replaced More precisely, the approximants are not imaginary translates of the truncated spectraloperator ζ ( ∂ ( T ) c ) but instead, they are the results of the Riemann zeta function applied (in thesense of the functional calculus) to imaginary translates of the truncated infinitesimal shifts ∂ ( T ) c ;namely, ζ ( ∂ ( T ) c + iτ ), for some τ ∈ R . The same cautionary comment as in the previous footnote applies here as well. by the truncated infinitesimal shifts ∂ ( T ) = ∂ ( T ) c , as is the case in § 7. Appendix A: On the Origins of Universality In 1885, Karl Weierstrass proved that the set of polynomials is dense (for thetopology of uniform convergence) in the space of continuous functions on a compactinterval of the real line. He also proved that the set of trigonometric polynomialsis dense (in the above sense) in the class of 2 π -periodic continuous functions on R . Several improvements of Weierstrass’ approximation theorem were obtained byBernstein (1912), M¨untz (1914), Wiener (1933), Akhiezer–Krein and Paley–Wiener(1934), as well as Stone (1947); (see [ PerQ ] and [ St1 ] for an interesting survey.)The first ‘universal’ object in mathematical analysis was discovered in 1914by Feteke. He showed the existence of a real-valued power series ∞ X n =1 a n x n (7.0.4)which is divergent for all real numbers x = 0. Moreover, this divergence is so extremethat, for every continuous function f on [ − , 1] such that f (0) = 0, there exists anincreasing sequence { N k } ∞ K =1 ⊂ N such thatlim k →∞ N k X n =1 a n x n = f ( x ) , (7.0.5)uniformly on [ − , f ( z ) such that, for every entire function g ( z ),there exists a sequence of complex numbers { a n } ∞ n =1 such thatlim n →∞ f ( z + a n ) = g ( z ) , (7.0.6)uniformly on all compact subsets of the complex plane.The term of ‘universality’ was used for the first time by J. Marcinkiewicz, whoobtained the following result:Let { h n } ∞ n =1 be a sequence of real numbers such that lim n →∞ h n = 0. Then,there exists a continuous function f ∈ C [0 , 1] such that, for every continuous func-tion g ∈ C [0 , { n k } ⊂ N such thatlim k →∞ f ( x + h n k ) − f ( x ) h n k = g ( x ) , (7.0.7)almost everywhere on [0 , f a universal primi-tive .It is in 1975 that S. M. Voronin discovered the first explicitly universal objectin mathematics, which is the Riemann zeta function ζ ( s ). His original universal-ity theorem (Theorem 2.1) states that any non-vanishing analytic function on the RUNCATED INFINITESIMAL SHIFTS AND QUANTIZED UNIVERSALITY OF ζ ( s ) 27 disk (of center and radius less than ) can be uniformly approximated by imagi-nary shifts of the Riemann zeta function. In the next appendix, we discuss variousextensions of this theorem. 8. Appendix B: On some Extensions of Voronin’s Theorem This appendix is dedicated to a discussion of some of the extensions ofVoronin’s theorem on the universality of the Riemann zeta function. (See, e.g.,[ Bag1, Bag2, Emi, GarSt, Gon, Lau1, Lau2, Lau3, LauMa1, LauMa2,LauMaSt, LauSlez, LauSt, KarVo, Tit, Wil, Rei1, Rei2, St1, St2 ].)Webegin by mentioning the first improvement of this theorem, which was obtained byBagchi and Reich in [ Bag1, Rei1 ], and then discuss further extensions to someother elements of the Selberg class of zeta functions. We refer the interested readerto J. Steuding’s monograph [ St1 ] for a detailed discussion of the historical devel-opments of the notion of universality in mathematics along with various extensionsof Voronin’s theorem to a large class of L -functions. The first im-provement of Voronin’s universality theorem was given independently by Bagchiand Reich in [ Bag1, Rei1 ]. These authors improved Voronin’s theorem by replac-ing the disk D (see Theorem 2.1) by any (suitable) compact subset of the righthalf of the critical strip (i.e., of { < Re ( s ) < } ). Their result, often referred toas the extended Voronin (or universality) theorem, has already been stated in § Remark . As was already mentioned in Remark 2.3, the condition accordingto which g ( s ) is non-vanishing is crucial and cannot be dropped. Indeed, it can beshown that if a function g ( s ) ( satisfying the conditions of Theorem 2.2 ) were tohave at least one zero ( i.e., if g ( s ) were to vanish somewhere in the compact subset K ) , then a contradiction to the Riemann hypothesis would be obtained; namely, thiswould imply the existence of a zero of ζ ( s ) which is not lying on the critical line ( see, e.g., [ KarVo, Lau1, St1 ]) . Moreover, if we take g ( s ) = ζ ( s ) , then the strip ofuniversality is the open right half of the critical strip; namely, { < Re ( s ) < } . Inother words, it is impossible to extend the universality property of the Riemannzeta function further inside the critical strip. Indeed, if such an extension existedon some region U , then U would have to intersect the critical line { Re ( s ) = } ,which ( by Hardy’s theorem, see [ Tit ]) contains infinitely many zeros of ζ ( s ) . Thiswould contradict the assumption according to which the target function ( i.e., ζ ( s )) does not have any zeros ( in the given compact set K ) . Remark . In the statement of the extended Voronin theorem ( Theorem 2.2 ) ,the compact set K is allowed to have empty interior. In that case, the function g ( s ) is allowed to be an arbitrary non-vanishing continuous function on K . Hence,taking K to be a compact subinterval of the real axis ( and taking into account someof the comments in Remark 8.1 ) , we conclude that any continuous curve can beapproximated by the Riemann zeta function ( and its imaginary translates ) . Furtherrefinements ( and an application of the extended universality theorem to ζ ( s ) itself ) enable one to see that the graph of ζ ( s ) contains arbitrary small “scaled copies”ofitself, a property characteristic of “fractality”. ( See [ Wil ] . )A variant of Voronin’s extended theorem (Theorem 2.2) about the universal-ity of ζ ( s ) was obtained by Reich [ Rei1, Rei2 ]. He restricted the approximatingshifts of a given target function to arithmetic progressions and obtained a ‘ discrete ’universality version of Voronin’s theorem. His result can be stated as follows: Theorem . Let K be a compact subset of the right critical strip { La5 , Appendix E], where many relevant references are also provided. See, e.g., [ La5 , Appendix C] and the many relevant references therein for the terminologyand definitions about modular forms which are used here. RUNCATED INFINITESIMAL SHIFTS AND QUANTIZED UNIVERSALITY OF ζ ( s ) 29 holomorphic ) in the interior of K . Then, given any ǫ > , there exists τ > suchthat sup s ∈ K | g ( s ) − L ( s + iτ, f ) | < ǫ. (8.2.1) Moreover, the set of admissible τ ’s is infinite and, in fact, lim inf T →∞ vol (cid:0) { τ ∈ [0 , T ] : sup s ∈ K | L ( s + iτ, f ) − g ( s ) | < ǫ } (cid:1) > . (8.2.2)An extension to Dirichlet series with multiplicative coefficients was obtainedby Laurincikas and Slezeviciene in [ LauSlez ].A number of additional results concerning the extensions and applicationsof the universality theorem to L -functions can be found in Steuding’s monograph[ St1 ]. Finally, we point out the fact that extensions to other families of zeta func-tions not necessarily belonging to the Selberg class of zeta functions (such as thefamily of Hurwitz zeta functions) were also obtained. Given α ∈ (0 , 1] and for Re ( s ) > 1, the Hurwitz zeta function is defined by ζ ( s, α ) = ∞ X m =0 m + α ) s . (8.2.3)This function has a meromorphic continuation to the whole complex plane. Ithas a simple pole at s = 1 with residue equal to 1. Note that for α = 1, we have ζ ( s, 1) = ζ ( s ), the Riemann zeta function, and that for α = , ζ ( s, ) is given (forall s ∈ C ) by ζ ( s, 12 ) = (2 s − ζ ( s ) . (8.2.4)Clearly, for α = 1 and α = , ζ ( s, α ) has an Euler product expansion. This is not the case, however, for α ∈ (0 , − { , } . As a result, except for α ∈ { , } , theHurwitz zeta function, defined by Equation (8.2.3), is not an element of the Selbergclass of zeta functions.An extension of Theorem 2.2 to the class of Hurwitz zeta functions for the case α ∈ (0 , −{ , } , where α is rational or transcendental , was obtained independentlyby Gonek in [ Gon ] and Bagchi in [ Bag1 ]. Their result can be stated as follows: Theorem . Suppose α ∈ (0 , − { , } is either rational or transcenden-tal . Let K ⊂ { < Re ( s ) < } be a compact subset with connected complement. Let g ( s ) be a continuous function on K which is analytic ( i.e., holomorphic ) in theinterior of K . Then, for every ǫ > , we have lim inf T → + ∞ vol (cid:0) { τ ∈ [0 , T ] : sup s ∈ K | g ( s ) − ζ ( s + iτ, α ) | < ǫ } (cid:1) > . (8.2.5) Remark . Note that in the statement of Theorem 8.7 and in contrast tothe case of the Riemann zeta function ( and other L -functions ) , the function g isallowed to have zeros inside the compact set K . This is not all that surprising since except for α ∈ { , } , the Hurwitz zeta function is notexpected to satisfy the Riemann hypothesis. In view of Equation (8.2.5), the Hurwitz zeta function ζ ( s, α ) can uniformlyapproximate target functions which may have zeros inside the compact subset K . Hence, the Hurwitz zeta function is an example of a mathematical object whichis ‘ strongly universal ’. The theory of strong universality has been developed in sev-eral directions. We note that the results obtained within our functional analyticframework about the truncated infinitesimal shifts ∂ ( T ) c and the truncated spectraloperators a ( T ) c = ζ ( ∂ ( T ) c ) can also be used to provide an operator-theoretic extensionof the notion of strong universality . 9. Appendix C: Almost Periodicity and the Riemann Hypothesis Let f be a holomorphic complex-valued function on some vertical strip S a, b = { s ∈ C : a < Re ( s ) < b } . Then, f is said to be almost periodic if for every ǫ > α , β such that a < α < β < b , there exists ℓ ( f, α, β, ǫ ) > t , t ) of length ℓ , there exists a number τ ∈ ( t , t ) such that forany α ≤ x ≤ β and any y ∈ R , we have | f ( x + iy + iτ ) − f ( x + iy ) | < ǫ. The notion of almost periodicity was introduced by H. Bohr in [ Boh1 ]. Heproved that any Dirichlet series is almost periodic in its half-plane of absoluteconvergence. Moreover, he showed in [ Boh2 ] that the almost periodicity of theclass of Dirichlet L -functions L ( s, χ ) with non-trivial character χ (i.e., χ = 1) isintimately connected with the location of the critical zeros of the Riemann zetafunction: Theorem . Given any character χ = 1 , then L ( s, χ ) is almost periodic inthe half-plane { Re ( s ) > } if and only if the Riemann hypothesis is true. Remark . The notion of almost periodicity was introduced by H. Bohr asan analytic tool for proving the Riemann hypothesis. We note that his approachfailed for the case of the Riemann zeta function but led to a reformulation of theRiemann hypothesis for the class of Dirichlet L -functions associated to a non-trivialcharacter. ( See Theorem 9.1 just above. ) In contrast to the Riemann zeta function,which has a pole at s = 1 and whose Dirichlet series and Euler product converge onlyfor Re ( s ) > , the Dirichlet L -functions with non-trivial characters are holomorphicfor Re ( s ) > even in all of C if the Dirichlet character χ is primitive ) andhave a Dirichlet series and an Euler product which converge for Re ( s ) > and inparticular, inside the critical strip : L ( s, χ ) = ∞ X n =1 χ ( n ) n s = Y p ∈P (1 − χ ( p ) p − s ) − , for Re ( s ) > . A key fact, also established by Bohr ( and central to his reformulation of RH ) is that ( still for a nontrivial character ) L ( s, χ ) is almost periodic in the criticalstrip ( and actually, for Re ( s ) > because its Euler product is convergent there. Recall that the Dirichlet L -function (or Dirichlet L -series) is initially defined by L ( s, χ ) := P ∞ n =1 χ ( n ) n s for Re ( s ) > 1; see, e.g., [ Pat ], [ Ser ]. The Euler product converges absolutely for Re ( s ) > Re ( s ) > 0; see,e.g., [ Ser ]. It is the convergence (and not the absolute convergence) which is essential here. RUNCATED INFINITESIMAL SHIFTS AND QUANTIZED UNIVERSALITY OF ζ ( s ) 31 Remark . The concept of almost periodicity is key to the proof and un-derstanding of the universality of ζ and of other L -functions. In fact, toward thebeginning of the 20th century, Bohr’s theory of almost periodicity was already usedby Harald Bohr and his collaborators in order to obtain several interesting resultsconcerning the Riemann zeta function ( and other Dirichlet L -functions ) , such asthe density of the range of ζ ( s ) along the vertical lines { Re ( s ) = c } , with < c < see Theorem 2.5 and the discussion preceding it in § ) . 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