Hardy spaces of Dirichlet Series and the Riemann Zeta Function Kristian Seip Norwegian University of Science and Technology NTNU Nonlinear Functional Analysis Valencia, October 20, 2017 1997 paper with Hedenmalm and Lindqvist In 1997, I published a paper with Håkan Hedenmalm and Peter Lindqvist in Duke Math. J. which begins with the following words: “The purpose of this paper is twofold. First we study systems of functions '(x),'(2x),'(3x),..., and second we study the Hardy space H2 of the infinite dimensional polydisk. Building on ideas of Arne Beurling and Harald Bohr, we find that the two topics are intimately connected, the common feature being the use of Dirichlet series.” Systems of dilated functions—an example More precisely, we were concerned with the case when ' is an odd 2-periodic function—the prototype being '(x) sin¼x. Our Æ interest in systems of the form '(nx) came from a nonlinear eigenvalue problem (for the one-dimensional p-Laplacian): d ³ p 2 ´ p 2 u0 ¡ u0 ¸ u ¡ u 0, dx j j Å j j Æ where 1 p , studied in depth by Ôtani in 1984. It has a Ç Ç 1 remarkable solution, and all eigenfunctions can be expressed in terms of dilations of a single odd periodic function, called sinp(x), an inverse of a certain Abelian integral, corresponding to the smallest eigenvalue ¸. Riesz bases of systems of dilated functions We were led from this special case to a general problem: Question When will a system of functions '(nx), n 1,2,3,..., ' odd and Æ 2-periodic, be a Riesz basis for L2(0,1)? When '(x) p2sin¼x, we have an orthonormal basis, and the Æ question is somehow how much a general ' can differ from p2sin¼x for the Riesz basis property to be “preserved”. (A problem well known from systems of exponentials!) An effective way of expressing analytically such “deviation” comes from work of Beurling (and also Wintner, as we later we discovered). The Beurling–Wintner transformation We may write X1 '(x) anp2sin(¼nx); Æ n 1 Æ then X1 X bn'(nx) cnp2sin(¼nx), (1) n 1 Æ n Æ P where cn d n anbn/d (which is called the Dirichlet convolution Æ j of (an) and (bn)). Therefore, the mapping X X1 s S anp2sin(¼nx): ann¡ n 1 Æ n 1 Æ Æ P s P s transforms (1) into a bona fide product of ann¡ and bnn¡ . This observation is found in the notes from a seminar held by Beurling in 1945, as well as in a 1944 paper of Wintner. The Hardy space of Dirichlet series H 2 Under S , L2(0,1) is mapped to the following space: Definition H 2 consists of all Dirichlet series of the form X1 s f (s) ann¡ , Æ n 1 Æ (s σ it a complex variable) with Æ Å X1 f 2 a 2 . H 2 n k k Æ n 1j j Ç 1 Æ 2 2 P 2σ Since by Cauchy–Schwarz, f (s) f 1 n , an f in H 2 n 1 ¡ 2 j j · k k Æ H is analytic in CÅ {s σ it : σ 1/2}. 1/2 Æ Æ Å È Observation on the Riesz basis problem By the Beurling–Wintner transformation, the following is essentially immediate from the definition of a Riesz basis: Observation (Hedenmalm–Lindqvist–Seip 97) The dilated system '(nx) is a Riesz basis for L2(0,1) if and only if both S ' and 1/S ' are multipliers for H 2. Recall: m a multiplier means m a Dirichlet series such that mf in H 2 whenever f is in H 2. (Then obviously m itself is in H 2 and f mf is a bounded operator.) 7! Reproducing kernel of H 2 Clearly, H 2 is a Hilbert space with reproducing kernel, and the kernel is Kw (s) ³(s w), Æ Å where ³ is the Riemann zeta-function defined in σ 1 by È X1 s Y s 1 ³(s) n¡ (1 p¡ )¡ . Æ n 1 Æ p ¡ Æ Thus, for f in H 2 and σ 1/2, we have È f (s) f ,³( s) . Æ h ¢ Å i Harald Bohr’s viewpoint s s s Set z1 2¡ , z2 3¡ , ..., zj pj¡ , ..., i.e. think of the Æ Æ s Æ “prime powers” p¡ as independent variables (by the fundamental theorem of arithmetic). 2 2 In this way, H may be thought of as H (D1). A simple estimate using the multiplicative definition of ³(s) 2 shows that the series converges in D1 ` .(CÅ can be \ 1/2 thought of as a subset.) The theorem on multipliers We denote the set of multipliers by M . As noted before, every multiplier m defines a bounded operator on H 2; the corresponding operator norm is denoted by m . k kM Theorem (Hedenmalm–Lindqvist–Seip 97) 2 M H H1(C0) and m M sup m(s) . Æ \ k k Æ s C0 j j 2 Thus the multipliers live a fully-fledged life in the larger half-plane C0. 2 We write M H 1 : H H1(C ). Æ Æ \ 0 Our proof uses the Bohr lift and relies on the fact that BH 1 H1(T1). Æ In hindsight: The multiplier theorem (stated and proved differently) was known to Toeplitz in the 1920s ... Beurling’s completeness problem: Cyclicity in H 2 By the Beurling–Wintner transformation, proving completeness of the system '(nx) in L2(0,1) amounts to proving cyclicity of S ' in H 2. This is a much more delicate problem, studied by Wintner, Beurling, Kozlov, Helson, Hedenmalm–Lindqvist–S, Nikolski (and possibly others). Only partial results are known. There is (not surprisingly), quantitatively, a huge difference between the two problems (Riesz basis and completeness). Example The Davenport series X1 sin(¼nx) ' (x): ¸ ¸ Æ n 1 n Æ yields a Riesz basis iff ¸ 1 and completeness iff ¸ 1/2. È È “In a shadow of RH” This metaphor is used by Nikolski (2012), in the title of so far the last significant paper on the dilation completeness problem, to illustrate the relation to the Riemann hypothesis (RH). Indeed, the following statement about dilations of a nonperiodic function is equivalent to RH. The Nyman–Báez-Duarte criterion Set ½(x): x [x] and '(x): ½(1/x). The function  (x) is in Æ ¡ Æ [0,1] the closure in L2(0, ) of the system 1 '(nx), n 1,2,3,... Æ A number of more direct nontrivial links to the Riemann zeta function and analytic number theory (not just “shadows”) appear in our subject. p P s H spaces of Dirichlet series n1 1 ann¡ Æ We follow Bayart (2002) and assume 0 p . Ç Ç 1 On the one hand, we define H p as the completion of the PN s set of Dirichlet polynomials P(s) ann¡ with respect Æ n 1 to the norm (quasi-norm when 0 p Æ1) Ç Ç µ Z T ¶1/p 1 p P H p lim P(it) dt . k k Æ T T 0 j j !1 s p On the other hand, via the Bohr lift zj pj ¡ , H is p Æ p isometrically isomorphic to H (T1), the closure in L (T1) of holomorphic polynomials in infinitely many variables (zj ). Explicitly, in multi-index notation we write n (p )®(n) and obtain Æ j X1 ®(n) Bf (z) anz , Æ n 1 Æ P s 2 P 2 the Bohr lift of f (s) 1 ann¡ . Note: f an . n 1 H 2 n Æ Æ k k Æ j j Overall goals We want to do the usual things: understand the basic function theoretic properties of H p study suitable analogues of classical operators acting on H p, typically operators defined in terms of symbols and the interplay between properties of the operator and function theoretic properties of the symbol and in addition to: understand how our theory of Hardy spaces H p interacts with relevant parts of number theory and possibly other fields. From one to infinitely many variables Our theorem on multipliers is an example of a result obtained by looking for analogies with the classical theory of Hardy spaces. It is in line with the perspective of Rudin’s classic treatise on complex analysis in polydiscs (from the preface): “Briefly, the object is to see how much of our extremely detailed knowledge about holomorphic functions in the unit disc (...) can be carried over to an analogous setting, namely to polydiscs.” This is natural and the most obvious approach we may take, and it has led to many interesting results of which only a few will be mentioned in this talk. There are however some less evident points that can be made about the relation to the classical theory of Hardy spaces. The Riemann zeta function—a reminder “Additive” definition: X1 s ³(s): n¡ , Re(s) 1. Æ n 1 È Æ “Multiplicative” definition (Euler product) 1 Y³ s´¡ ³(s): 1 p¡ , Re(s) 1. Æ p ¡ È The multiplicative representation is what relates ³(s) to ¼(x) (the number of primes p x), for instance by the formula · Z 1 ¼(x) log³(s) s s dx. Æ 2 x(x 1) ¡ The motivation for the study of ³(s) is our desire to understand the behavior of ¼(x). Interplay between additive and multiplicative structure The interplay between the additive and the multiplicative embodied in ³(s) appears in our context in the following way: Interplay between function theory in half-planes (“additive” model) and function theory on polydiscs (“multiplicative” model) Another version of this which is sometimes useful to keep in mind: the dual group (in the Pontryagin sense) of T1 is the multiplicative group Q which has as well an additive structure (the order inducedÅ by the numerical size of positive rational numbers).