A HILBERT SPACE OF DIRICHLET SERIES AND

SYSTEMS OF DILATED FUNCTIONS IN L^{2}(0, 1)

Håkan Hedenmalm, Peter Lindqvist, and Kristian Seip

Abstract. For a function φ in L^{2}(0, 1), extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates φ(nx), n = 1, 2, 3, …, constitutes a Riesz basis or a complete sequence in L^{2}(0, 1). The problem translates into a question concerning multipliers and cyclic vectors in the Hilbert space H of Dirichlet series f(s) = ∑_{n} a_{n}n^{−s}, where the coefficients a_{n} are square summable. It proves useful to model H as the H^{2} space of the infinite-dimensional polydisk, or, which is the same, the H^{2} space of the character space, where a character is a multiplicative homomorphism from the positive integers to the unit circle. For given f in H and characters χ, f_{χ}(s) = ∑_{n} a_{n}χ(n)n^{−s} is a vertical limit function of f. We study certain probabilistic properties of these vertical limit functions.

1. Introduction

The purpose of this paper is twofold. First, we study systems of functions of the form φ(x), φ(2x), φ(3x), ..., and second, we consider the Hardy space H^{2} 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.

Let φ ∈ L^{2}(0, 1) be given and consider φ as defined on the whole real axis by extending it to an odd periodic function of period 2. The Riesz-Fischer theorem of Fourier analysis states that for φ(x) = \sqrt{2} sin(πx) the sequence φ(nx), n = 1, 2, 3, ..., is an orthonormal basis in the Hilbert space L^{2}(0, 1). The question raised in this paper is which functions can take the place of the sine in this theorem. It is clear that the statement must be weakened, because the only orthogonal bases are obtained from φ(x) = C sin(πx). If we instead ask for a classification of those φ for which the system {φ(nx)}_{n} is a Riesz basis (a basis orthonormal with respect to an equivalent norm) or of those φ for which the same system is a complete sequence in L^{2}(0, 1), we are led to profound problems.

The latter of the two problems – the completeness problem – was stated by Beurling in his seminar on harmonic analysis in Uppsala in 1945. A brief note from this seminar is found in [1]. Beurling’s note indicates that a natural way to

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The work of the first author was supported in part by the Swedish Natural Science Research Council (NFR), and that of the second and third authors by the Norwegian Research Council (NFR). While at MSRI, Berkeley, the first and third authors were also supported in part by NSF grant DMS-9022140.

approach these problems is to associate to the given function

φ(x) = ∑_{n=1}^{∞} a_{n}\sqrt{2} sin(nπx)

the Dirichlet series

Sφ(s) = ∑_{n=1}^{∞} a_{n}n^{−s},  (1-1)

and to try to express the Riesz basis and completeness properties in terms of analytic properties of Sφ(s). This approach has proved fruitful. We have solved completely the Riesz basis problem (Theorems 3.1 and 5.2 below): the system {φ(nx)}_{n} is a Riesz basis in L^{2}(0, 1) if and only if (1-1) defines an analytic function bounded away from zero and infinity in the half-plane ℜs > 0.

A major part of this paper consists of a study of Dirichlet series of the form (1-1) with ∑_{n} |a_{n}|^{2} < +∞. Such Dirichlet series form, in a natural way, a Hilbert space of analytic functions on the half-plane ℜs > \frac{1}{2}. From now on, we denote this space by H. In Sections 2–4, we seek to reveal the basic analytic features of this space. A central problem is to characterize the so-called multipliers of H. Theorem 3.1 below states that the multipliers are precisely the bounded analytic functions in the right half-plane ℜs > 0 which can be represented as Dirichlet series. This result is crucial for the characterization of the Riesz bases {φ(nx)}_{n} mentioned above. Apart for its relevance for the dilation Riesz basis problem, the study of the space H can also be motivated by the mere fact that its kernel function K_{H}(z, w) is more or less the Riemann zeta function: K_{H}(z, w) = ζ(z + w̄).

Following an idea of Bohr, we find it both convenient and illuminating to use the infinite-dimensional polydisk for the study of H. This leads us to power series in infinitely many variables, a concept studied already by Hilbert, and to an identification of H as the Hardy space H^{2} of the infinite-dimensional polydisk. The infinitedimensional polycircle can be identified with the group Ξ of complex-valued characters χ on the positive integers, which satisfy |χ(n)| = 1 and χ(mn) = χ(m)χ(n). The characters constitute the (compact) dual group of the discrete multiplicative group of positive rationals ⟨ℚ_{+}, ·⟩.

A function f ∈ H is said to be cyclic provided that the collection of functions fg, where g is a finite Dirichlet series, is dense in H. A basic observation is that the system {φ_{n}}_{n} is complete if and only if the Dirichlet series Sφ is cyclic in H. We restate cyclicity in terms of our model of H as the H^{2} space of the infinitedimensional polydisk, and state some conditions for cyclicity. There is some overlap between these conditions and results due to Henry Helson [16], who studied cyclicity in a more general context.

Since the multipliers of H extend analytically and boundedly to ℜs > 0, but the functions of H need only be analytic on ℜs > \frac{1}{2}, one is led to suspect that, nevertheless, in a sense to be made precise, a function in H, picked at random, almost surely extends holomorphically to ℜs > 0. Given a Dirichlet series in H,

f(s) = ∑_{n=1}^{∞} a_{n}n^{−s},

DIRICHLET SERIES AND SYSTEMS OF DILATED FUNCTIONS IN L^{2}(0, 1) 3

we consider, for characters χ,

f_{χ}(s) = ∑_{n=1}^{∞} a_{n}χ(n)n^{−s},

which again is an element of H. The functions f_{χ} are the normal limits of vertical translates of f. Almost every one of them has a convergent Dirichlet series on ℜs > 0, and is hence holomorphic there. The ”almost every” is with respect to the Haar measure on the character group Ξ, which is normalized to have total mass 1. This result was obtained in a more abstract setting by Helson [16]. A curious consequence is that a probabilistic “Riemann hypothesis” (Corollary 4.8) holds: for almost all characters χ, the Dirichlet series ζ_{χ}(s) = ∑_{n=1}^{∞} χ(n)n^{−s} converges to a zero-free holomorphic function in the region ℜs > \frac{1}{2}. The convergence part has the following interpretation: for almost all characters χ,

∑_{n=1}^{N} χ(n) = O(N^{1/2+ε}) as N → +∞

holds for fixed ε > 0. For each n, it is fruitful to regard the function χ(n) as a stochastic variable, which for n > 1 is uniformly distributed on 𝕋. The stochastic variables χ(p), where p runs through the primes, are then mutually independent, and the mutual dependence of the χ(n), as n runs through the positive integers, is governed by the multiplicative rule χ(mn) = χ(m)χ(n). The above estimate of the partial sum ∑_{n:n≤N} χ(n) is what one would expect if all the χ(n) were mutually independent, so although they do satisfy complicated multiplicative dependence relations, this is insignificant additively.

The above-mentioned assertions may be compared with Jean-Pierre Kahane’s paper [18] on random Dirichlet series. Kahane works with functions f_{χ}(s), where χ(n) is treated as a sequence of independent random variables, and hence no multiplicativity property holds.

In [15], Helson suggests that the classical theory of Dirichlet series be combined with modern techniques from harmonic and functional analysis. We hope that the present paper can inspire work in that direction.

2. The Hilbert spaces H and H^{2}(𝔻^{∞})

2.1 The space H and preliminaries on Dirichlet series. We are concerned with Dirichlet series of the form

f(s) = ∑_{n=1}^{∞} a_{n}n^{−s},  (2-1)

where s = σ + it is a complex variable. We recall briefly some classical facts about such series. There are a number of critical lines or abscissæ connected with (2-1). We have the abscissa of absolute convergence σ_{a} and the abscissa of ordinary convergence σ_{c}. These numbers are such that the series converges in the prescribed sense to the right but not to the left of the abscissa in question. We also have the abscissa of uniform convergence σ_{u}, defined as the infimum of those σ_{0} for which the series converges uniformly in the half-plane ℜs > σ_{0}. We have trivially −∞ ≤ σ_{c} ≤ σ_{u} ≤ σ_{a} ≤ +∞ and σ_{a} − σ_{c} ≤ 1 if anyone of the abscissæ is finite. A theorem of Bohr [4] says that σ_{a} − σ_{u} ≤ \frac{1}{2}, and this inequality is sharp, as was shown by Bohnenblust and Hille [2].

When we need to indicate which function we are taking the abscissæ of, we do this by putting the function in parenthesis; for instance, we would write σ_{c}(f) for the abscissa of convergence. In terms of the partial sums of the coefficients, S_{N} = ∑_{n:n≤N} a_{n}, the abscissa of convergence of the Dirichlet series (2-1) can be expressed as follows: unless σ_{c}(f) is negative, σ_{c}(f) equals the infimum of all positive real numbers α for which S_{N} = O(N^{α}) as N → +∞.

Of relevance to us is also the abscissa of regularity and boundedness σ_{b}, which is the infimum of those σ_{0} for which the function defined by (2-1) (possibly by analytic continuation from a smaller half-plane) is analytic and bounded for ℜs > σ_{0}. We shall need the following result of Bohr [3].

Lemma 2.1. (Bohr’s theorem) σ_{u} = σ_{b}.

For a more complete account of the basic facts on the convergence of Dirichlet series, the reader is referred to [12] and Bohr’s thesis [5].

We will assume that the function f given by the Dirichlet series (2-1) belongs to H, that is,

∑_{n=1}^{∞} |a_{n}|^{2} < +∞.  (2-2)

The Cauchy-Schwarz inequality yields

( ∑_{n=1}^{∞} |a_{n}n^{−s}|)^{2} ≤ ∑_{n=1}^{∞} |a_{n}|^{2} ∑_{n=1}^{∞} n^{−2σ}, s = σ + it,

and so the abscissa of absolute convergence is at most \frac{1}{2} for the series (2-1). The abscissa of convergence may equal \frac{1}{2}, as is seen by the example a_{n} = (n^{1/2} log(n + 1))^{−1}.

As mentioned in the introduction, the collection of Dirichlet series (2-1) satisfying (2-2) is denoted by H. It is a complex Hilbert space when endowed with the inner product

⟨f, g⟩_{H} = ∑_{n=1}^{∞} a_{n}b̄_{n},

where f(s) = ∑_{n} a_{n}n^{−s}, and g(s) = ∑_{n} b_{n}n^{−s}. Thus, formally, H is just l^{2}(ℕ), where ℕ = {1, 2, 3, … } is the collection of natural numbers. However, as a Hilbert space of analytic functions in the half-plane ℜs > \frac{1}{2}, H has a rich and interesting structure. To be specific, let us mention the classical problem of multiplication of two Dirichlet series. Formally, the product of f(s) = ∑_{n} a_{n}n^{−s} and g(s) = ∑_{n} b_{n}n^{−s} is again a Dirichlet series

f(s)g(s) = ∑_{n} c_{n}n^{−s},  (2-3)

with

c_{n} = ∑_{k,l:kl=n} a_{k}b_{l}.

For f ∈ H and g ∈ H, the product fg need not be in H, although the abscissa of absolute convergence of (2-3) can be shown to be at most \frac{1}{2} (as it is for functions in H). We are lead to the multiplier problem: Find those functions m, analytic in ℜs > \frac{1}{2}, for which m(s)f(s) is in H for every f ∈ H. The collection of these multipliers m is denoted by M. Theorem 3.1 below solves the multiplier problem.

Let us finally mention a classical theorem which we will refer to from time to time. We say that a set of real numbers ξ_{1}, ξ_{2}, ..., ξ_{r} is ℚ-linearly independent if

c_{1}ξ_{1} + c_{2}ξ_{2} + ⋯ + c_{r}ξ_{r} = 0

holds with integer coefficients c_{1}, c_{2}, ..., c_{r} only when the coefficients are all zero. By the fundamental theorem of arithmetic, the prime example of a ℚ-linearly independent set of numbers is the image under the logarithm function of any finite set of different primes. We have the fundamental

Lemma 2.2. (Kronecker’s theorem) Suppose the real numbers 1, ϑ_{1}, ϑ_{2}, ..., ϑ_{k} are ℚ-linearly independent. Let α_{1}, α_{2}, …, α_{k} be arbitrary real numbers, and N and ϵ be given positive. Then there are integers

n > N, q_{1}, q_{2}, ..., q_{k}

such that

|nϑ_{m} − q_{m} − α_{m}| < ϵ (m = 1, 2, ..., k).

For a proof, we refer to Chapter XXIII of [13].

To give an example of how Kronecker’s theorem applies to our context, we mention that an immediate consequence is the following identity:

∥1 + ∑_{p} a_{p}p^{−s}∥_{∞} = 1 + ∑_{p} |a_{p}|;

here the sum is only over prime indices, and

∥f∥_{∞} = sup_{ℜs>0} |f(s)|.  (2-4)

2.2. The Hardy space H^{2} of the infinite-dimensional polydisk. Let f be the Dirichlet series (2-1), and suppose f ∈ H. A fundamental observation, first made by Bohr [4], is that if we put

z_{1} = 2^{−s}, z_{2} = 3^{−s}, …, z_{m} = p_{m}^{−s}, …,

where p_{m} denotes the m-th prime, then, in view of the fundamental theorem of arithmetic, the Dirichlet series (2-1) can be considered as a power series of infinitely many variables. The point is, as Bohr clarifies in his work, that the quantities z_{m} = p_{m}^{−s}, m = 1, 2, 3, …, “obwohl sie ja in Wirklichkeit Funktionen nur des einen Parameters s sind, sich in mancher Beziehung fast ganz benehmen, als wären sie von einander unabhängige Variable.” We put z = (z_{m})_{m} = (z_{1}, z_{2}, z_{3}, …), and write

n = p_{k_{1}}^{ν_{1}}p_{k_{2}}^{ν_{2}} … p_{k_{r}}^{ν_{r}},

so that (2-1) formally takes the form

𝔔f(z) = ∑_{n=1}^{∞} a_{n}z_{k_{1}}^{ν_{1}} · z_{k_{2}}^{ν_{2}} ⋯ z_{k_{r}}^{ν_{r}}.  (2-5)

From now on, for a given element f ∈ H, 𝔔f denotes the corresponding power series, and we drop the relationship between z and s.

Another, sometimes more convenient, way to think of the extension 𝔔f of f in (2-5) is to write it as

𝔔f(ϕ) = ∑_{n=1}^{∞} a_{n}ϕ(n),  (2-5’)

where ϕ is a quasi-character, by which we mean a mapping ϕ : ℕ → ℂ, which is multiplicative, ϕ(mn) = ϕ(m)ϕ(n), and has ϕ(1) = 1 and ϕ(n) ∈ 𝔻 for n > 1. Here, 𝔻 is the open unit disk. If we put, for the m-th prime p_{m}, ϕ(p_{m}) = z_{m}, and identify ϕ with z = (z_{m})_{m}, then (2-5) and (2-5’) express the same function 𝔔f. The Cauchy-Schwarz inequality applied to (2-5) (or (2-5’)) and Euler’s product formula give

|𝔔f(z)|^{2} = |𝔔f(ϕ)|^{2} ≤ ∑_{n=1}^{∞} |a_{n}|^{2} ∑_{n=1}^{∞} |ϕ(n)|^{2}

= ∥f∥_{H}^{2} · ∏_{p}(1 − |ϕ(p)|^{2})^{−1} = ∥f∥_{H}^{2} · ∏_{m=1}^{∞}(1 − |z_{m}|^{2})^{−1},

provided that |z_{m}| < 1 for every m. It follows that we have bounded point evaluation at z, id est,

|𝔔f(z)| ≤ C(z) ∥f∥_{H},

if and only if |z_{m}| < 1 for every m and

∑_{m=1}^{∞} |z_{m}|^{2} < +∞.

We denote by 𝔻^{∞} the set of z = (z_{m})_{m=1}^{∞} for which |z_{m}| < 1 for every m, and call it the infinite-dimensional polydisk. The collection of power series (2-5) with f ∈ H can be regarded as a space of analytic functions on 𝔻^{∞} ∩ l^{2}(ℕ). We denote this space by H^{2}(𝔻^{∞}), and supply it with the Hilbert space structure induced by H. Note that H^{2}(𝔻^{∞}) differs from H only in that its elements are considered as functions on 𝔻^{∞} ∩ l^{2}(ℕ) rather than on the half-plane ℜs > \frac{1}{2}. In other words, H results from considering (2-5’) for the special quasi-characters ϕ_{s}(n) = n^{−s}.

Before discussing further the nature of H^{2}(𝔻^{∞}), we mention an interesting problem, treated already by Hilbert [17]: Is it possible to extend the meaning of (2-5) beyond the set 𝔻^{∞} ∩ l^{2}(ℕ), at least under certain favorable cirumstances? As indicated by Bohr, Hilbert’s solution to this problem has a remarkable significance for Dirichlet series [4]. Let

Ω = {z = (z_{m})_{m} ∈ 𝔻^{∞} : |z_{m}| < β_{m} for every m},

where {β_{m}}_{m} is a sequence of positive numbers, 0 < β_{m} ≤ 1. For a given point z = (z_{1}, z_{2}, …) ∈ 𝔻^{∞}, we put

z^{(m)} = (z_{1}, z_{2}, …, z_{m}, 0, 0, …),

id est, the j-th coordinate for j > m is put equal to zero. For the time being, let f be a Dirichlet series (2-1), about which we assume only that it converges on some half-plane ℜs > σ_{c} (we do not require (2-2) to hold). The formal power series 𝔔f(z^{(m)}) is the “m-te Abschnitt” of 𝔔f at z, and it only depends on the finitely many coordinates z_{1}, …, z_{m}. If the “m-te Abschnitt” converges absolutely on Ω for each m, and there exists a constant C = C(f) such that

|𝔔f(z^{(m)})| ≤ C, z ∈ Ω,

independently of m, 𝔔f is said to be finitely bounded in Ω. In particular, the collection of series 𝔔f that are finitely bounded in 𝔻^{∞} is denoted by H^{∞}(𝔻^{∞}), and we write

∥𝔔f∥_{H^{∞}(𝔻^{∞})} = sup_{m,z} |𝔔f(z^{(m)})|,

where z ranges over 𝔻^{∞}. It is natural to define the value of 𝔔f(z) as the limit of 𝔔f(z^{(m)}) as m → +∞. The limit makes sense for z ∈ 𝔻^{∞} ∩ c_{0}(ℕ), where c_{0}(ℕ) is the closed subspace of l^{∞}(ℕ) of sequences that converge to 0. Moreover, the function 𝔔f(z) is a bounded analytic function on 𝔻^{∞} ∩ c_{0}(ℕ). Indeed, by the Schwarz lemma, we have, for n < m,

|𝔔f(z^{(m)}) − 𝔔f(z^{(n)})| ≤ 2 ∥𝔔f∥_{H^{∞}(𝔻^{∞})} max {|z_{j}| : n < j ≤ m}.

For z ∈ 𝔻^{∞} ∩ c_{0}(ℕ), the maximum on the right hand side tends to zero as both n and m tend to infinity, so that the above estimate says that the 𝔔f(z^{(m)}) form a Cauchy sequence in the space of bounded analytic functions on 𝔻^{∞} ∩ c_{0}(ℕ). We note that the supremum of |𝔔f(z)| on 𝔻^{∞} ∩ c_{0}(ℕ) agrees with the norm of f in H^{∞}(𝔻^{∞}).

It follows from our proof of the multiplier theorem in the next section that H^{∞}(𝔻^{∞}) corresponds to the set of Dirichlet series representing bounded analytic functions in ℜs > 0; in fact

∥f∥_{∞} = ∥𝔔f∥_{H^{∞}(𝔻^{∞})}.  (2-6)

This norm identity reflects in a concise way Bohr’s observation that the p^{−s} behave as “independent variables”.

Thus far, we have thought about the space H^{2}(𝔻^{∞}) as the Hardy space H^{2} on the infinite-dimensional polydisk 𝔻^{∞}. A perhaps more natural setting is to regard the spaces H^{2}(𝔻^{∞}) and H^{∞}(𝔻^{∞}) as function spaces on the distinguished boundary 𝕋^{∞}. To this end, we need a group-theoretical identification of 𝕋^{∞}, which we shall now describe. Suppose that χ : ℕ → ℂ satisfies

(i) χ(mn) = χ(m)χ(n), m, n = 1, 2, 3, … (ii) |χ(n)| = 1.

We say that χ is a character, and write χ ∈ Ξ. We tacitly understand that χ(n^{−1}) = χ(n)^{−1}, so that the multiplicative formula holds for all positive rational numbers ℚ_{+}. The characters χ constitute the dual group of ⟨ℚ_{+}, ·⟩ (ℚ_{+} is given the discrete topology, so that the space Ξ of characters is compact). Examples of characters are the unit character χ(n) ≡ 1, and, more generally, for t ∈ ℝ,

χ(n) = n^{it} = e^{it log n}.

The space Ξ can be identified with the infinite-dimensional polycircle 𝕋^{∞} in the following way. Given a point z = (z_{1}, z_{2}, …) ∈ 𝕋^{∞}, we define the value of χ at the primes through

χ(2) = z_{1}, χ(3) = z_{2}, …, χ(p_{m}) = z_{m}, …,

and extend the definition multiplicatively. This then yields a character, and clearly, all characters are obtained by this procedure. This provides a natural identification of the character space Ξ with 𝕋^{∞}. The product topology on 𝕋^{∞} makes it a compact space, and it is easily shown that it corresponds to the topology Ξ gets as a dual group, so that the identification Ξ = 𝕋^{∞} is topological. There exists a unique (apart from normalization) Haar measure on Ξ, which we identify with the ordinary product measure ρ on 𝕋^{∞}. More precisely, let λ be the arc length measure on 𝕋, normalized so that λ(𝕋) = 1, and put, for Borel subsets E_{1}, E_{2}, …, E_{N} of 𝕋, and E = E_{1} × E_{2} × … × E_{N} × 𝕋 × 𝕋 × … ⊂ 𝕋^{∞},

ρ(E) = λ(E_{1})λ(E_{2}) · . . . · λ(E_{N});

this defines a Borel measure ρ on 𝕋^{∞}, which coincides with the Haar measure on Ξ, once it is agreed that the Haar measure should have total mass 1. We shall think of ρ as living on Ξ as well as on 𝕋^{∞}.

We return to the setting of the function f ∈ H given by (2-1). The series

∑_{n=1}^{∞} a_{n}χ(n)  (2-7)

converges in the norm of L^{2}(Ξ, ρ) (though we do not know if we have pointwise convergence almost everywhere) to a function 𝔓f(χ), which is determined uniquely by the two requirements that it be in L^{2}(Ξ, ρ) and that

∫_{Ξ} χ̄(q)𝔓f(χ) dρ(χ) = a_{q}, q ∈ ℚ_{+},

where we declare a_{q} = 0 for q ∈ ℚ_{+} \ ℕ. This function 𝔓f is an extension to the characters of the earlier defined function 𝔔f on the quasi-characters. Our shorthand for the above relationship between the function 𝔓f(χ) and the coefficients {a_{n}}_{n} will be

𝔓f(χ) ∼ ∑_{n} a_{n}χ(n), χ ∈ Ξ.

It is clear that the Plancherel identity

∫_{Ξ} |𝔓f|^{2}dρ = ∑_{n=1}^{∞} |a_{n}|^{2} = ∥f∥_{H}^{2}

holds.

Let H^{2}(Ξ) (also written as H^{2}(𝕋^{∞})) be the closed subspace of L^{2}(Ξ, ρ) consisting of functions that can be expanded in a series (2-7), with {a_{n}}_{n} ∈ l^{2}(ℕ). Furthermore, let H^{∞}(Ξ) be the intersection L^{∞}(Ξ, ρ) ∩ H^{2}(Ξ), which is a closed subspace of L^{∞}(Ξ, ρ). To get the connection between the Hardy spaces on the distinguished boundary 𝕋^{∞} and on the interior 𝔻^{∞}, we need the operator 𝔈 : H^{2}(𝕋^{∞}) → H^{2}(𝔻^{∞}), which associates with a function g(χ) ∼ ∑_{n} b_{n}χ(n) in H^{2}(Ξ) the function 𝔈g(ϕ) = ∑_{n} b_{n}ϕ(n) for quasi-characters ϕ ∈ 𝔻^{∞} ∩ l^{2}(ℕ). It is clear that it is an isometric isomorphism. The operators 𝔔, 𝔓, and 𝔈 are related via 𝔔 = 𝔈𝔓.

Lemma 2.3. The operators 𝔓 : H → H^{2}(Ξ) and 𝔈 : H^{2}(Ξ) → H^{2}(𝔻^{∞}) are isometric isomorphisms. Furthermore, the restriction of 𝔈 to H^{∞}(Ξ) is an isometric isomorphism H^{∞}(Ξ) → H^{∞}(𝔻^{∞}).

Proof. The first part just restates what was already done above. For the rest, we can refer to [8], where it is shown that the space H^{∞}(ρ) is canonically isometrically isomorphic to both H^{∞} on 𝔻^{∞} ∩c_{0}(ℕ) and on 𝔻^{∞} ∩l^{2}(ℕ). In [8], H^{∞}(ρ) is defined as the weak-star closure in L^{∞}(𝕋^{∞}, ρ) of the infinite-dimensional polydisk algebra A(𝔻^{∞}). We have the inclusion H^{∞}(ρ) ⊂ H^{2}(𝕋^{∞}), and by inspection, the above canonical mapping coincides with our 𝔈. It follows that it suffices to check that the spaces H^{∞}(ρ) and H^{∞}(𝕋^{∞}) are the same as (closed) subspaces of L^{∞}(Ξ, ρ). Since both H^{∞}(ρ) and H^{∞}(𝕋^{∞}) are subspaces of H^{2}(𝕋^{∞}), it is enough to check that their images under 𝔈 coincide. In [8], it was shown that the operator 𝔈 was in fact furnished by integrating against the Poisson kernel in each variable, so that 𝔈 applied to H^{∞}(𝕋^{∞}) consists of bounded analytic functions on 𝔻^{∞} ∩ l^{2}(ℕ). By the above-mentioned result from [8], the assertion of the lemma follows. □

2.3. Vertical limit functions. The notion of characters allows us to clarify an important property of H. A natural unitary operator on H is that of vertical translation,

T_{τ} f(s) = f(s + iτ).

Fix an f ∈ H. To every sequence τ_{n} of translations there exists a subsequence, say τ_{n(k)}, such that T_{τ_{n(k)}}f(s) converges uniformly on compact subsets of the domain ℜs > \frac{1}{2} to a limit function, say f̃(s). We will say that f̃ is a vertical limit function of f. We have the following result.

Lemma 2.4. The vertical limit functions of the function f ∈ H given by (2-1) coincide with the functions of the form

f_{χ}(s) = ∑_{n=1}^{∞} a_{n}χ(n)n^{−s},  (2-8)

χ being a character.

Proof. It is clear that every vertical limit function is of the form (2-8). The fact that every function of this form is a vertical limit function is a consequence of Kronecker’s theorem (Lemma 2.2). □

3. The theorem on multipliers

We put

ℂ_{+} = {s = σ + it ∈ ℂ : σ > 0},

and let, as usual, H^{∞}(ℂ_{+}) denote the set of bounded analytic functions on ℂ_{+}. By D we denote the collection of functions f which can be represented by a convergent Dirichlet series

f(s) = ∑_{n=1}^{∞} a_{n}n^{−s}

in some half-plane σ_{c} < ℜs < +∞. A multiplier m on H is by definition a holomorphic function on the half-plane ℜs > \frac{1}{2} with the property that mf ∈ H whenever f ∈ H. By standard functional analysis, the operation of multiplying by a multiplier is a bounded operator on H, and the multipliers form a commutative Banach algebra. The collection of all multipliers on H is denoted by M. We find it convenient to consider a multiplier both as a function and as a bounded linear operator on H. The operator norm of m is denoted by ∥m∥_{M} and the supremum norm in ℂ_{+} by ∥m∥_{∞}.

Our theorem on multipliers can be stated as follows.

Theorem 3.1. We have M = D ∩ H^{∞}(ℂ_{+}). Moreover, ∥m∥_{M} = ∥m∥_{∞} holds for m ∈ M.

The proof of Theorem 3.1 splits into two parts.

3.1. Proof that M ⊂ D ∩ H^{∞}(ℂ_{+}) and ∥m∥_{∞} ≤ ∥m∥_{M}. Let m ∈ M be given. The idea of the proof is to show that m lifts to a multiplier on H^{2}(𝔻^{∞}), from which we deduce that it belongs to H^{∞}(𝔻^{∞}). This then entails that m is a Dirichlet series, and that it is bounded throughout ℂ_{+}. Since 1 ∈ H, we have m ∈ H, and hence

m(s) = ∑_{n=1}^{∞} b_{n}n^{−s}, ℜs > \frac{1}{2},

where the sequence {b_{n}}_{n} is in l^{2}(ℕ). So, if we apply the operator 𝔓 to m and f, where f ∈ H, we get that both 𝔓m and 𝔓f are in H^{2}(Ξ). Consequently, their product 𝔓m 𝔓f is in L^{1}(Ξ, ρ). Since m is a multiplier, we have mf ∈ H, so that 𝔓(mf) ∈ H^{2}(Ξ). We wish to prove that 𝔓(mf) = 𝔓m 𝔓f as functions in L^{1}(Ξ, ρ). To this end, note that for finite Dirichlet series f, this is verified by direct calculation. The general case when f ∈ H is arbitrary then follows by approximating f with finite Dirichlet series.

Since ∥mf∥_{H} ≤ ∥m∥_{M}∥f∥_{H}, we get, by successively plugging in f = 1, m, m^{2}, and so on, that ∥m^{j}∥_{H} ≤ ∥m∥_{M}^{j} holds for j = 1, 2, . . .. By what we just did, 𝔓(m^{j}) = (𝔓m)^{j}, so that

(∫_{Ξ} |𝔓m|^{2j}dρ)^{1/(2j)} = ∥(𝔓m)^{j}∥_{H^{2}(Ξ)}^{1/j} = ∥m^{j}∥_{H}^{1/j} ≤ ∥m∥_{M}.

As j → +∞, the left hand side tends to ∥𝔓m∥_{L^{∞}(Ξ,ρ)}. We conclude that 𝔓m belongs to L^{∞}(Ξ, ρ) as well as H^{2}(Ξ), that is, 𝔓m ∈ H^{∞}(Ξ). It also follows that ∥𝔓m∥_{H^{∞}(Ξ)} ≤ ∥m∥_{M}. By Lemma 2.3, the function 𝔔m = 𝔈𝔓m is in H^{∞}(𝔻^{∞}), and its norm equals the norm of 𝔓m in H^{∞}(Ξ). Recall that by Hilbert’s approach to power series of infinitely many variables, the functions in the space H^{∞}(𝔻^{∞}) are bounded and analytic in 𝔻^{∞} ∩ c_{0}(ℕ), and that the formula giving 𝔔m on (part of) the quasi-characters is

𝔔m(ϕ) = ∑_{n=1}^{∞} b_{n}ϕ(n), ϕ ∈ 𝔻^{∞} ∩ l^{2}(ℕ).

Plugging in the special quasi-character ϕ_{s}(n) = n^{−s}, we get our function m(s) back: m(s) = 𝔔m(ϕ_{s}) for ℜs > \frac{1}{2}. Since ϕ_{s} depends analytically on the parameter s and is in 𝔻^{∞} ∩ c_{0}(ℕ) for ℜs > 0, the fact that 𝔔m is bounded and holomorphic on 𝔻^{∞} ∩ c_{0}(ℕ) implies that m(s) is bounded and analytic in ℂ_{+}, and ∥m∥_{∞} ≤ ∥𝔔m∥_{H^{∞}(𝔻^{∞})} = ∥𝔓m∥_{H^{∞}(Ξ)} ≤ ∥m∥_{M}.

3.2. Proof that D ∩ H^{∞}(ℂ_{+}) ⊂ M and ∥m∥_{M} ≤ ∥m∥_{∞}. The key to the proof of the converse relation is the following lemma, which is due to Fritz Carlson [7].

Lemma 3.2. (Carlson’s theorem) Let f(s) = ∑_{n=1}^{∞} a_{n}n^{−s} be convergent (and hence analytic) in ℂ_{+} and bounded in every half-plane ℜ(s) > δ with δ > 0. Then, for each σ > 0,

∑_{n=1}^{∞} |a_{n}|^{2}n^{−2σ} = lim_{T →+∞} \frac{1}{2T} ∫_{−T}^{T} |f(σ + it)|^{2}dt.

An immediate corollary is the following.

Lemma 3.3. If f(s) = ∑_{n=1}^{∞} a_{n}n^{−s} is convergent and bounded in ℂ_{+}, then f ∈ H and

∥f∥_{H} = lim_{σ→0^{+}} ( lim_{T →+∞} \frac{1}{2T} ∫_{−T}^{T} |f(σ + it)|^{2}dt)^{1/2} .

Remark. Note that together with Lemma 2.1, Lemma 3.3 implies Bohr’s inequality σ_{a} − σ_{u} ≤ \frac{1}{2}.

We carry on with the proof, and suppose that

m(s) = ∑_{n=1}^{∞} b_{n}n^{−s}

converges in some half-plane ℜs > σ_{0}, and that it extends boundedly and holomorphically to ℂ_{+}. Then, by Bohr’s theorem (Lemma 2.1), the Dirichlet series defining m(s) actually converges uniformly to m(s) on every half-plane ℜs > ε, with ε > 0. Let f ∈ H have Dirichlet series ∑_{n=1}^{∞} a_{n}n^{−s}, and introduce, for N = 1, 2, 3, …, the cut-off series

f_{N}(s) = ∑_{n=1}^{N} a_{n}n^{−s}, s ∈ ℂ.

The function mf_{N} is given by a convergent Dirichlet series in ℂ_{+}, and it is bounded there. We can now apply Lemma 3.3 to the function mf_{N}, to obtain

∥mf_{N}∥_{H} ≤ ∥m∥_{∞}∥f_{N}∥_{H} ≤ ∥m∥_{∞}∥f∥_{H}.

Since ∥mf∥_{H} ≤ sup_{N} ∥mf_{N}∥_{H}, it follows that

∥mf∥_{H} ≤ ∥m∥_{∞}∥f∥_{H},

which completes the proof of Theorem 3.1.

Remark. In the proof of Theorem 3.1, we actually prove that the multipliers on H may be identified with the space H^{∞}(𝔻^{∞}). It is remarkable that just being able to extend the function m ∈ H holomorphically and boundedly to ℂ_{+} should entail that the function 𝔔m ∈ H^{2}(𝔻^{∞}) is bounded on 𝔻^{∞} ∩ l^{2}(ℕ) (and hence bounded and analytic on 𝔻^{∞} ∩ c_{0}(ℕ)). After all, the condition on m just corresponds to the behavior of 𝔔m along the one-dimensional complex variety which is the image of ℂ_{+} under s ↦ ϕ_{s}.

4. Some function theoretic properties of H

We first recall some results from ergodic theory. We then turn to the almost sure behavior of the vertical limit functions f_{χ}, with χ ∈ Ξ, of a given f ∈ H. As an application, we consider the famous zeta function, which here plays the role of the kernel function. After that, we look at the function-theoretic properties of individual functions in H; in particular, we study zero sets.

4.1. Preliminaries from ergodic theory: Kronecker flows. Given a collection of real numbers α_{1}, α_{2}, α_{3}, …, we consider the continuous group of vertical translations

T_{t}(z_{1}, z_{2}, z_{3}, …) = (e^{−itα_{1}}z_{1}, e^{−itα_{2}}z_{2}, e^{−itα_{3}}z_{3}, …),

acting on the infinite-dimensional polycircle 𝕋^{∞}, where t ranges over the reals. The Kronecker flow {T_{t}}_{t} is known to be ergodic if and only if for each fixed n, n = 1, 2, 3, …, the numbers α_{1}, α_{2}, …, α_{n} are ℚ-linearly independent. This is done for finite-dimensional polycircles 𝕋^{n} in [9, pp. 64, 67, 69, 99]. After a few minor modifications, the proof in [9], which is based on the approximation property in Kronecker’s theorem (Lemma 2.2), covers the infinite-dimensional case as well. We pick α_{j} = log p_{j}, where p_{j} denotes the j-th prime, and note that by the fundamental theorem of arithmetic, any finite subset of the collection {α_{j}}_{j} is ℚ-linearly independent, so that the flow {T_{t}}_{t} is ergodic. If we write out the flow explicitly, we get

T_{t}(z_{1}, z_{2}, z_{3}, …) = (2^{−it}z_{1}, 3^{−it}z_{2}, 5^{−it}z_{3}, …), t ∈ ℝ.

As in Section 2, points z = (z_{1}, z_{2}, …) in 𝕋^{∞} are identified with elements χ of the character group Ξ by putting χ(p_{j}) = z_{j} for the j-th prime p_{j}. The flow then takes the more elegant form

(T_{t}χ)(n) = n^{−it}χ(n), n ∈ ℕ, t ∈ ℝ.

We note that the ergodicity of the flow {T_{t}}_{t} may now be checked off directly from condition (iv) in [9, p. 99]. By the Birkhoff-Khinchin ergodic theorem [9, pp. 11–12, 39, 99], we have

lim_{T →+∞} \frac{1}{2T} ∫_{−T}^{T} g(T_{t}χ_{0}) dt = ∫_{Ξ} g(χ) dρ(χ)  (4-1)

for every χ_{0} ∈ Ξ if g is continuous on Ξ, and for almost every χ_{0} if we only assume that g ∈ L^{1}(Ξ, ρ). We now apply this result in the context of our space H. For f ∈ H, with series expansion f(s) = ∑_{n} a_{n}n^{−s}, we write f_{σ}(s) = f(σ + s) = ∑_{n} a_{n}n^{−σ−s}, and note that for σ ≥ 0, this is again an element of H. Recall that σ_{b}(f) ∈ [−∞, \frac{1}{2}] is the abscissa of boundedness for f, and that by Bohr’s theorem (Lemma 2.1), it coincides with the abscissa of uniform convergence. It follows that for σ, σ_{b}(f) < σ < +∞, the Dirichlet series for f_{σ}(s) is uniformly convergent on ℜs > 0, so that by Kronecker’s theorem (Lemma 2.2), the partial sums of the series 𝔓f_{σ} converge uniformly on Ξ, making 𝔓f_{σ} continuous on Ξ. Now, by (4-1),

lim_{T →+∞} \frac{1}{2T} ∫_{−T}^{T} |𝔓f_{σ}(T_{t}χ_{0})|^{2} dt = ∫_{Ξ} |𝔓f_{σ}(χ)|^{2} dρ(χ) = ∑_{n=1}^{∞} |a_{n}|^{2}n^{−2σ}  (4-2)

holds for almost all χ_{0} ∈ Ξ if 0 ≤ σ ≤ σ_{b}(f), and for all χ_{0} (in particular for the unit character χ_{0} ≡ 1) if σ > \frac{1}{2}. Notice the close resemblance with Carlson’s theorem (Lemma 3.2). Indeed, Carlson’s theorem can be read off from (4-2), with χ_{0} ≡ 1.

4.2. The almost sure behavior of vertical limit functions. As before, let f ∈ H be given by (2-1), and write f_{χ} for the vertical limit function of f, given by (2-8). The function f_{χ} shares with f the property that it is holomorphic on ℜs > \frac{1}{2}. Helson [16] has shown that the function f_{χ} extends analytically to ℜs > 0 and that its Dirichlet series converges there, for almost every χ ∈ Ξ. We wish to illuminate his argument, and obtain additional properties of the vertical limit functions. To make the statement as precise as possible, we need the space H_{i}^{2}(ℂ_{+}) of functions f holomorphic in ℂ_{+} which have f ∘ φ ∈ H^{2}(𝔻); here H^{2}(𝔻) is the usual Hardy space of the unit disk 𝔻, and φ is the Cayley transform φ(z) = (1 − z)/(1 + z). The subscript “i” stands for (conformal) invariance. Let λ_{i} be the probability measure dλ_{i}(t) = π^{−1}(1 + t^{2})^{−1}, which is obtained as the image under the Cayley transform of the normalized arc length measure on the unit circle:

∫_{−∞}^{+∞} |f(it)|^{2}dλ_{i}(t) = \frac{1}{2π} ∫_{−π}^{π} |f ∘ φ(e^{iθ})|^{2}dθ, f ∈ H_{i}^{2}(ℂ_{+}).

The space H^{2}(𝔻) is usually regarded as a subspace of L^{2}(𝕋), and likewise the space H_{i}^{2}(ℂ_{+}) may be considered to be a subspace of L^{2}(iℝ, λ_{i}^{∗}), the space of functions g on iℝ that have g(it) in L^{2}(ℝ, λ_{i}).

Apart from the ergodic statement, the following theorem is due to Helson [14]. For the benefit of the reader, we adapt Helson’s proof to the present setting.

Theorem 4.1. Let f ∈ H be given, with series expansion (2-1), and let Π be a countable collection of absolutely continuous Borel probability measures on the real line. For almost every character χ ∈ Ξ, the function

f_{χ}(s) = ∑_{n=1}^{∞} a_{n}χ(n)n^{−s}, ℜs > \frac{1}{2},

extends analytically to an element of H_{i}^{2}(ℂ_{+}), has

∫_{−∞}^{+∞} |f_{χ}(it)|^{2}dϖ(t) < +∞, for all ϖ ∈ Π,

and enjoys

\frac{1}{2T} ∫_{−T}^{T} |f_{χ}(it)|^{2}dt → ∑_{n=1}^{∞} |a_{n}|^{2} as T → +∞.

Proof. We first decide on how to define f_{χ}(it). It should correspond to the possibly divergent sum ∑_{n} a_{n}χ(n)n^{−it}. This sum, however, makes sense as 𝔓f(T_{t}χ), for almost every χ, where {T_{t}}_{t} is the ergodic Kronecker flow of the previous subsection. We thus put f_{χ}(it) = 𝔓f(T_{t}χ), and observe that by (4-2), with σ = 0, this function is locally in L^{2} along the imaginary axis, and has the asserted property

\frac{1}{2T} ∫_{−T}^{T} |f_{χ}(it)|^{2}dt → ∑_{n=1}^{∞} |a_{n}|^{2}  as T → +∞,  (4-4)

almost surely in χ. Moreover, by Fubini’s theorem, for a Borel probability measure ϖ,

∫_{Ξ} ∫_{−∞}^{+∞} |f_{χ}(it)|^{2}dϖ(t) dρ(χ) = ∫_{−∞}^{+∞} ∫_{Ξ} |f_{χ}(it)|^{2}dρ(χ) dϖ(t)

= ∫_{−∞}^{+∞} ∥f∥_{H}^{2} dϖ(t) = ∥f∥_{H}^{2} < +∞.

In particular, the function f_{χ}(it), considered as a function of t, almost surely is square integrable on the real line with respect to ϖ. Elementary measure theory shows that the same holds true simultaneously for all measures ϖ in Π, since the latter set is countable. By the same token, f_{χ}(it) is almost surely in L^{2}(ℝ, λ_{i}).

A function g in L^{2}(𝕋) is in H^{2}(𝔻) if and only if

∫_{𝕋} z^{n}g(z) dλ(z) = 0, n = 1, 2, 3, …,

where λ is as before the normalized arc length measure on the unit circle. After an application of the Cayley transform, we have that a function g ∈ L^{2}(iℝ, λ_{i}^{∗}) is in H_{i}^{2}(ℂ_{+}) if and only if

∫_{−∞}^{+∞} (\frac{1 − it}{1 + it})^{n} g(it) dλ_{i}(t) = 0, n = 1, 2, 3, ….

Therefore, to see that f_{χ} is almost surely in H_{i}^{2}(ℂ_{+}), it suffices to check that

∫_{−∞}^{+∞} (\frac{1 − it}{1 + it})^{n} f_{χ}(it) dλ_{i}(t) = 0, n = 1, 2, 3, …,

for almost all χ. However, to check that an L^{2}(Ξ, ρ) function vanishes almost everywhere, it is enough to show that all of its Fourier coefficients are 0. We set a_{q} = 0 for q ∈ ℚ_{+} \ ℕ, and integrate the left hand side of the above expression against χ̄(q) to get the Fourier coefficients:

∫_{Ξ} χ̄(q) ∫_{−∞}^{+∞} (\frac{1−it}{1+it})^{n} f_{χ}(it) dλ_{i}(t)

= ∫_{−∞}^{+∞} (\frac{1−it}{1+it})^{n} ∫_{Ξ} χ̄(q) f_{χ}(it) dρ(χ) dλ_{i}(t) = ∫_{−∞}^{+∞} (\frac{1−it}{1+it})^{n} a_{q}q^{−it} dλ_{i}(t) = 0,

where we have used Fubini’s theorem, and that a_{q} = 0 for q < 1. This completes the proof. □

The ergodic reasoning behind (4-4) leads to an estimate of f_{χ}, which does not seem to follow from Helson’s work.

Theorem 4.2. Let f and f_{χ} be as in Theorem 4.1, and write s = σ + it. Then almost surely in χ ∈ Ξ,

|f_{χ}(s) − a_{1}| ≤ C (\frac{1 + |t|^{1/2}}{σ^{1/2}} ) , s ∈ ℂ_{+},

for some constant C = C(f, χ), 0 < C < +∞. Moreover, almost surely in χ,

f_{χ}(s) = a_{1} + o (\frac{|t|^{1/2}}{σ^{1/2}} ) as |t| → +∞

holds uniformly in σ > 0.

Proof. As in (4-4), we have, by ergodic theory, that almost surely in χ,

\frac{1}{T} ∫_{0}^{T} (f_{χ}(iτ) − a_{1})^{2}dτ → 0 as |T | → +∞,

the space average of the function (in χ) being 0. This entails that the function

F_{χ}(ix) = ∫_{0}^{x} (f_{χ}(iτ) − a_{1})^{2}dτ

meets

F_{χ}(ix) = o(|x|),  as |x| → +∞,  (4-5)

almost surely in χ. By Theorem 4.1, the function f_{χ} is almost surely in H_{i}^{2}(ℂ_{+}), so that the squared function (f_{χ} − a_{1})^{2} almost surely belongs to the analogously defined space H_{i}^{1}(ℂ_{+}). As such it is given by the Poisson formula

(f_{χ}(σ + it) − a_{1})^{2} = \frac{σ}{π} ∫_{−∞}^{+∞} \frac{(f_{χ}(ix) − a_{1})^{2}}{σ^{2} + (t − x)^{2}} dx.

Integrating by parts, using (4-5), we obtain the representation formula

(f_{χ}(σ + it) − a_{1})^{2} = \frac{2σ}{π} ∫_{−∞}^{+∞} \frac{t − x}{(σ^{2} + (t − x)^{2})^{2}} F_{χ}(ix) dx,

for σ > 0 and t ∈ ℝ. After an application of the size control (4-5) to this integral, the desired estimates follow by taking square roots. □

For a certain class of functions in H, the estimate of Theorem 4.2 can be improved considerably. Note that the conclusion is that the growth in the imaginary direction is precisely what the Schnee-Landau theorem requires to imply convergence of Dirichlet series [5, 20].

Corollary 4.3. Let f ∈ H be such that all powers f^{N}, where N is a positive integer, also are in H. Then, for each ε > 0, we have, almost surely in χ ∈ Ξ,

|f_{χ}(s) − a_{1}| ≤ C (\frac{1 + |t|^{ε}}{σ^{ε}} ) , s = σ + it ∈ ℂ_{+},

for some constant C = C(χ, f, ε), 0 < C < +∞.

Proof. The estimate is more or less immediate from Theorem 4.2. □

The general question about almost sure convergence of the Dirichlet series of vertical limit functions was treated by Helson in [16], in a somewhat more general context. He obtained the following basic result.

Theorem 4.4. (Helson) Let f ∈ H be given, with series expansion (2-1). For almost every character χ, the Dirichlet series

f_{χ}(s) = ∑_{n} a_{n}χ(n)n^{−s}

converges in the half-plane ℜs > 0.

According to a classical formula for the abscissa of convergence [12, pp. 6–8] (see also Section 2 of the present paper), Theorem 4.4 is equivalent to the statement that, almost surely in χ, the partial sum function

S_{N}(χ) = ∑_{n:n≤N} a_{n}χ(n)  (4-6)

has S_{N}(χ) = O(N^{ε}) as N → +∞, almost surely in χ, for all ε > 0. So, in light of Helson’s theorem, we find the following special case interesting.

Theorem 4.5. Let f ∈ H be given, with series expansion (2-1). Suppose a_{n} = 0 for all composite numbers. For almost every character χ, we then have S_{N}(χ) = O(1) as N → ∞, where S_{N}(χ) is as in (4-6).

Proof. The non-composite numbers are the primes and unity (1). Without loss of generality, we may suppose that a_{1} = 0. As the parameter p runs through the primes, the χ(p), treated as functions of χ, run through the distinct coordinate variables in the polycircle 𝕋^{∞}, which is our standard realization of the character space Ξ. This has a clear interpretation: the random variables

χ ↦ a_{p}χ(p), p = 2, 3, 5, 7, 11, ...

are mutually independent, have mean squares 0, and variances σ_{p}^{2} = |a_{p}|^{2}.

For a given positive integer N and a positive real number M, let E(N, M) be the set of all characters χ for which |S_{N}(χ)| < M. By Kolmogorov’s inequality [9, p. 260], we have

ρ( ∩_{N} E(N, M)) ≥ 1 − M^{−2} ∑_{p} σ_{p}^{2} = 1 − M^{−2}|a_{p}|^{2}.

By letting M tend to +∞, we see that S_{N}(χ) = O(1) as N → +∞ holds for almost all χ ∈ Ξ. □

4.3. The kernel function. Riemann’s zeta function for random characters. Given a separable Hilbert space A of analytic functions on a domain Ω in ℂ or in ℂ^{n}, one forms the kernel function K_{A}(z, w) by taking some orthonormal basis {e_{n}(z)}_{n} in A, and putting

K_{A}(z, w) = ∑_{n} e_{n}(z) ē_{n}(w), (z, w) ∈ Ω × Ω.

This then proves to be independent of the particular choice of orthonormal basis, and has the reproducing property that

f(w) = ⟨f, K_{A}(·, w)⟩_{A}, w ∈ Ω.

In fact, as an element of A, K(·, w) is uniquely determined by its reproducing property. In H, an orthonormal basis is supplied by e_{n}(z) = n^{−z}, for n = 1, 2, 3, …, so that its kernel function is

K_{H}(z, w) = ∑_{n=1}^{∞} n^{−z−w̄} = ζ(z + w̄), ℜz > \frac{1}{2}, ℜw > \frac{1}{2},

where ζ(s) is the Riemann zeta function:

ζ(s) = ∑_{n=1}^{∞} n^{−s}, ℜs > 1.

In Section 2, we modeled the space H as both the Hardy space on the infinitedimensional polydisk 𝔻^{∞} and the polycircle 𝕋^{∞}. The polycircle 𝕋^{∞} was identified with the character group Ξ of the multiplicative positive rationals. In particular, for quasi-characters ϕ ∈ 𝔻^{∞} ∩ l^{2}(ℕ), the ”point evaluation” f ↦ 𝔔f(ϕ) is a continuous linear functional on H, so that it too must be given by a kernel function K_{H^{2}(𝔻^{∞})}(ψ, ϕ),

𝔔f(ϕ) = ⟨𝔔f, K_{H^{2}(𝔻^{∞})}(·, ϕ)⟩_{H^{2}(𝔻^{∞})}, ϕ ∈ 𝔻^{∞} ∩ l^{2}(ℕ).

In terms of a series expansion, it is written

K_{H^{2}(𝔻^{∞})}(ψ, ϕ) = ∑_{n=1}^{∞} ψ(n) ϕ̄(n), ψ, ϕ ∈ 𝔻^{∞} ∩ l^{2}(ℕ).

The inner product in H, and hence in H^{2}(𝔻^{∞}), is better visualized on the distinguished boundary 𝕋^{∞} = Ξ, where we have

⟨f, g⟩_{H} = ⟨𝔔f, 𝔔g⟩_{H^{2}(𝔻^{∞})} = ⟨𝔓f, 𝔓g⟩_{H^{2}(𝕋^{∞})}

= ∫_{Ξ} 𝔓f(χ) \overline{𝔓g}(χ) dρ(χ), f, g ∈ H.

This suggests introducing the kernel

K_{H^{2}(Ξ)}(χ, ϕ) = ∑_{n=1}^{∞} χ(n) ϕ̄(n), χ ∈ Ξ, ϕ ∈ 𝔻^{∞} ∩ l^{2}(ℕ),

where the sum, for fixed ϕ, is understood to converge in the sense of the space L^{2}(Ξ, ρ). This kernel then has the reproducing property

𝔔f(ϕ) = ⟨𝔓f, K_{H^{2}(Ξ)}(·, ϕ)⟩_{H^{2}(Ξ)}

= ∫_{Ξ} 𝔓f(χ) \overline{K_{H^{2}(Ξ)}}(χ, ϕ) dρ(χ), ϕ ∈ 𝔻^{∞} ∩ l^{2}(ℕ).

On the other hand, it is well known that the reproducing kernel on H^{2}(𝕋^{N}) for finite-dimensional polydisks is given as the product of the Cauchy kernel in each variable,

K_{H^{2}(𝕋^{N})}(z, w) = ∏_{n=1}^{N}(1 − w̄_{n}z_{n})^{−1}, z ∈ 𝕋^{N}, w ∈ 𝔻^{N},

and in the limit as the dimension N tends to infinity, we get [8]

K_{H^{2}(𝕋^{∞})}(χ, ϕ) = ∏_{n=1}^{∞}(1 − w̄_{n}z_{n})^{−1}

= ∏_{p} (1 − ϕ̄(p)χ(p))^{−1}, χ ∈ 𝕋^{∞}, ϕ ∈ 𝔻^{∞} ∩ l^{2}(ℕ),

where the second product runs over the primes, and z_{n} = χ(p_{n}), w_{n} = ϕ(p_{n}), for the n-th prime p_{n}. For ϕ ∈ 𝔻^{∞} ∩ l^{1}(ℕ), the above product converges pointwise to a continuous function of χ, but in general we must interpret the above product as being convergent in L^{2}(Ξ, ρ) [8] (Cole and Gamelin used martingale theory to obtain the convergence). Since the kernel function of a given Hilbert space of analytic functions is unique, we arrive at the equality K_{H^{2}(Ξ)} = K_{H^{2}(𝕋^{∞})}, that is, the Euler identity

∑_{n=1}^{∞} χ(n) ϕ̄(n) = ∏_{p} (1 − χ(p)ϕ̄(p))^{−1}, χ ∈ 𝕋^{∞}, ϕ ∈ 𝔻^{∞} ∩ l^{2}(ℕ),

which holds pointwise in χ for ϕ ∈ 𝔻^{∞} ∩ l^{1}(ℕ), and almost everywhere for general ϕ. This is a surprising interpretation of the Euler identity as arising from two ways of looking at one and the same kernel function! If we specialize to the particular quasi-characters ϕ(n) = n^{−s̄}, we get the more familiar

∑_{n=1}^{∞} χ(n) n^{−s} = ∏_{p} (1 − χ(p)p^{−s})^{−1},  χ ∈ Ξ, ℜs > \frac{1}{2},  (4-7)

with pointwise convergence in χ for ℜs > 1. We shall write ζ_{χ}(s) for the analytic function on ℜs > 1 given by either side of (4-7). By Lemma 2.4, suitably modified, these are the vertical limit functions of the zeta function ζ(s).

Theorem 4.6. Suppose the coefficients {a_{n}}_{n} are totally multiplicative and square summable, with a_{1} = 1. Then, for almost every character χ, the Dirichlet series

f_{χ}(s) = ∑_{n=1}^{∞} a_{n}χ(n)n^{−s}, ℜs > \frac{1}{2},

converges to a zero-free analytic function on the half-plane ℜs > 0.

Proof. The convergence statement follows from Theorem 4.4. We check that the function f_{χ}(s) almost surely lacks zeros in ℜs > 0. Let μ(n) be the Möbius function, which has μ(1) = 1, μ(n) = (−1)^{k} if n is the product of k different primes, and μ(n) = 0 if n is divisible by a square (other than 1). The Möbius function enables us to express the reciprocal of f_{χ}(s),

1/f_{χ}(s) = ∑_{n=1}^{∞} μ(n)a_{n}χ(n)n^{−s}, ℜs > 1,

because the coefficients {a_{n}}_{n} are totally multiplicative. By Theorem 4.1, 1/f_{χ}(s) extends analytically to ℂ_{+} almost surely in χ, whence the assertion follows. □

Theorem 4.6 has a curious interpretation for the vertical limit functions ζ_{χ}(s) of the Riemann zeta function: the Riemann hypothesis holds for almost every χ (see also [16]).

Corollary 4.7. (Helson) For almost every character χ, the Dirichlet series ζ_{χ}(s) = ∑_{n} χ(n)n^{−s} converges on the half-plane ℜs > \frac{1}{2} to an analytic function which has no zeros there.

Proof. Apply Theorems 4.1 and 4.6 to the coefficients a_{n} = n^{− \frac{1}{2} −ε}, with ε > 0. The assertion follows, but only in the slightly smaller half-plane ℜs > \frac{1}{2} + ε. An elementary measure-theoretical argument now yields the desired result. □

By the formula for the abscissa of convergence of a Dirichlet series (see Section 2), Corollary 4.7 has the following consequence.

Corollary 4.8. Let ε > 0. For almost every character χ, we have ∑_{n=1}^{N} χ(n) = O(N^{1/2+ε}) as N → +∞.

Question. Can the above be sharpened to the statement that, almost surely, ∑_{n=1}^{N} χ(n) = O(\sqrt{N} log N) as N → +∞?

Using Corollary 4.3, we can control the growth of the logarithm of ζ_{χ}(s). The branch of the logarithm intended is the one with

log ζ_{χ}(s) = − ∑_{p} log (1 − χ(p)p^{−s}), ℜs > 1,

where the logarithms on the right hand side are given by the principal branch.

Theorem 4.9. Suppose the coefficients {a_{n}}_{n} are totally multiplicative and square summable, with a_{1} = 1. Then, for almost every character χ, the function f_{χ}(s) = ∑_{n} a_{n}χ(n)n^{−s} has a logarithm

log f_{χ}(s) = − ∑_{p} log (1 − a_{p}χ(p)p^{−s}), ℜs > \frac{1}{2},

(the right hand side involves the principal branch) which extends holomorphically to ℜs > 0, and enjoys (s = σ + it)

log f_{χ}(s) = o (log \frac{|t|}{σ}) as |t| → +∞,

uniformly in σ, σ > 0.

Proof. Let α be a complex parameter. We define the power

f(s)^{α} = ∏_{p} (1 − a_{p}p^{−s})^{−α}, ℜs > \frac{1}{2},

where the right hand side employs the principal branch of the logarithm. The identification of H with H^{2}(Ξ) in Section 2 allows us to express the square of the norm of f^{α} in H as

∏_{p} \frac{1}{2π} ∫_{−π}^{π} |(1 − a_{p}e^{iθ})^{−α}|^{2}dθ.

Using a Maclaurin expansion of (1 − a_{p}e^{iθ})^{−α}, one sees that for α confined to a compact subset of ℂ,

\frac{1}{2π} ∫_{−π}^{π} |(1 − a_{p}e^{iθ})^{−α}|^{2}dθ = 1 + O(|a_{p}|^{2}).

It follows that f^{α} ∈ H. In particular, f^{γN} ∈ H for all N = 1, 2, 3, …, and γ = 1, −1, i, −i. By Corollary 4.3, we have, for each ε > 0, and almost all χ,

(f_{χ}(s))^{γ} = O (\frac{|t|^{ε}}{σ^{ε}} ) as |t| → +∞,

uniformly in σ, σ > 0, whence the assertion follows. □

Corollary 4.10. Fix ε > 0. Almost surely in χ,

log ζ_{χ}(σ + it) = o(log |t|) as |t| → +∞

holds uniformly in σ, σ > \frac{1}{2} + ε.

Proof. Apply Theorem 4.9 to the coefficients a_{n} = n^{− \frac{1}{2} −δ}, with δ = \frac{1}{2}ε. □

4.5. Function-theoretic properties of individual functions in H. The purpose of this paragraph is to obtain some basic information about the function theoretic properties of individual functions in H. In particular, we are interested in the structure of their zero sequences. The situation starkly contrasts the ”almost sure” behavior that we have concentrated on so far, because here, the functions are only known to be analytic in the half-plane ℜs > \frac{1}{2} (which we denote by ℂ_{\frac{1}{2}} ), and zeros may actually accumulate at a boundary point such as s = \frac{1}{2}.

Let H_{∞}^{2}(ℂ_{\frac{1}{2}} ) denote the uniformly local H^{2} space on ℂ_{\frac{1}{2}} : a function g holomorphic on ℂ_{\frac{1}{2}} is said to be in it if

sup_{θ∈ℝ} sup_{σ>1/2} ∫_{θ}^{θ+1} |g(σ + it)|^{2}dt < +∞.

It is a Banach space, and the functions in it are bounded in every half-plane ℜs > σ_{0}, with σ_{0} > \frac{1}{2}.

Theorem 4.11. We have the inclusion H ⊂ H_{∞}^{2}(ℂ_{\frac{1}{2}} ), and the injection mapping is continuous.

Proof. Let f ∈ H have the series expansion (2-1). We wish to prove that for all σ > \frac{1}{2},

∫_{θ}^{θ+1} |f(σ + it)|^{2}dt ≤ C ∑_{n=1}^{∞} |a_{n}|^{2}  (4-8)

holds, where C is an absolute constant. We note that it suffices to obtain (4-8) for finite Dirichlet series f, since on compact subsets of ℂ_{\frac{1}{2}} , elements of H are uniformly approximable by them. Moreover, by making a vertical translation, we see that we can set θ = 0, and by the Poisson integral formula, we see that it suffices to consider the limit case σ = \frac{1}{2}. By duality, we have

∫_{0}^{1} |f(\frac{1}{2} + it)|^{2}dt = sup_{g} | ∫_{0}^{1} ∑_{n} a_{n}n^{−1/2}e^{−it log n}g(t) dt|^{2}

= sup_{g} | ∑_{n} a_{n}n^{−1/2}ĝ(log n)|^{2} ≤ ∑_{n=1}^{∞} |a_{n}|^{2} sup_{g} ∑_{n=1}^{∞} n^{−1}|ĝ(log n)|^{2}, (4-9)

where the supremum is taken over all g ∈ L^{2}(0, 1) of norm ≤ 1, and ĝ is the Fourier transform of g,

ĝ(ξ) = ∫_{0}^{1} e^{−itξ}g(t) dt,

which extends to an entire function of exponential type ≤ 1. For such functions ĝ ,

∑_{n=1}^{∞} n^{−1}|ĝ(log n)|^{2} ≤ C ∫_{−∞}^{+∞} |ĝ(ξ)|^{2}dξ

holds for some absolute constant C. This follows from a suitable adaptation of a classical inequality of Plancherel and Pólya [21, pp. 96–98]. Heuristically speaking, the reason is that the left hand side looks like a Riemann sum of part of the integral to the right, and that the functions ĝ are sufficiently smooth. Modulo the Plancherel identity, the assertion now follows from (4-9). □

Corollary 4.12. Let f ∈ H, f(s) ≢ 0. If {s_{k}}_{k}, s_{k} = σ_{k} + it_{k}, denotes the sequence of zeros of f in ℂ_{\frac{1}{2}} , then

sup_{θ∈ℝ} ∑_{k:s_{k}∈Q_{θ}}(σ_{k} − \frac{1}{2}) < +∞,

where Q_{θ} is the half-strip ℜs > 0, θ < ℑs < θ + 1.

Proof. By Theorem 4.11, f(s)/s is contained in H^{2}(ℂ_{\frac{1}{2}} ), and so \{s_{k}\}_{k} satisfies the Blaschke condition

∑_{k} \frac{σ_{k} − \frac{1}{2}}{1 + |s_{k}|^{2}} < +∞.

All convergent nontrivial Dirichlet series have a zero-free half-plane, so that

sup_{k} σ_{k} < +∞.

Hence

A(θ) = ∑_{k:s_{k}∈Q_{θ}}(σ_{k} − \frac{1}{2}) < +∞,

for each real θ. If the function A(θ) were unbounded, we could find a sequence θ_{k} of reals such that A(θ_{k}) → +∞. But then the functions f_{k}(s) = f(s + iθ_{k}) would tend to 0 uniformly on compact subsets of ℂ_{\frac{1}{2}} , because they would have an ever increasing mass of zeros, and their norms in H_{∞}^{2}(ℂ_{\frac{1}{2}} ) are uniformly bounded. By Lemma 2.4, any vertical limit function has the form

∑_{n=1}^{∞} a_{n}χ(n)n^{−s},

where χ is a character. Such a function cannot be identically zero. This is a contradiction, and so A(θ) is bounded. □

Proposition 4.13. There exists a function f ∈ H, other than the 0 function, whose zero set contains a subsequence tending to s = \frac{1}{2} along the real line.

Proof. Let {b_{n}}_{n} be the sequence b_{1} = 0, b_{n} = n^{−1/2}(log n)^{−1} for n > 1, and notice that it has ∑_{n} b_{n}^{2} < +∞ and ∑_{n} n^{−1/2}b_{n} = +∞. We now select integers n_{k} and points σ_{k} according to the following fashion. Let n_{0} = 1. Given n_{k}, we pick σ_{k} > \frac{1}{2} so close to \frac{1}{2} and n_{k+1} so much larger than n_{k} that

∑_{n=n_{k}}^{n_{k+1}−1} n^{−σ_{k}}b_{n} − ∑_{n=1}^{n_{k}−1} n^{−σ_{k}}b_{n} − ∑_{n=n_{k+1}}^{∞} n^{−σ_{k}}b_{n} > 0.

The sequence \{σ_{k}\}_{k} is strictly decreasing, with limit \frac{1}{2}. If we put a_{n} = (−1)^{k}b_{n} for n_{k} ≤ n < n_{k+1}, and let f be as in (2-1), then f is real-valued on the real half-axis ]\frac{1}{2}, +∞[, and the sign of f(σ_{k}) alternates as (−1)^{k}. By the intermediate value theorem of calculus, f has infinitely many zeros along the real line, which accumulate at s = \frac{1}{2}. □

5. Systems of dilated functions in L^{2}(0, 1)

In this section, we return to the basis and completeness problems described in the introduction.

Let φ ∈ L^{2}(0, 1) be given and again consider it as an odd periodic function of period 2. The basic questions are: For which φ is the system

φ(x), φ(2x), φ(3x), …

a Riesz basis in L^{2}(0, 1), and for which φ is the same system complete in L^{2}(0, 1)? We recall that completeness means that the system spans a dense linear subspace.

The observation that the only orthogonal bases of this kind are generated by the functions φ(x) = C sin(πx) motivates introducing the representation

φ(x) = ∑_{n=1}^{∞} a_{n}\sqrt{2} sin(nπx)

for an arbitrary φ ∈ L^{2}(0, 1), and seeking solutions in terms of the coefficients a_{n}. For both problems, a necessary condition is a_{1} ≠ 0, because otherwise the function sin(πx) cannot be approximated. It is no restriction to normalize: a_{1} = 1 from now on.

Let \{e_{n}\}_{n} be the standard orthonormal basis e_{n}(x) = \sqrt{2} sin(nπx). Following Beurling [1], we associate to each f ∈ L^{2}(0, 1) with sine series expansion f(x) = ∑_{n} c_{n}e_{n}(x) the auxiliary Dirichlet series

Sf(s) = ∑_{n=1}^{∞} c_{n}n^{−s}.

Let ∥ · ∥_{L^{2}} and ⟨·, ·⟩_{L^{2}} be the norm and inner product in L^{2}(0, 1), respectively. The operator S is an isometric isomorphism between L^{2}(0, 1) and H: ∥Sf∥_{H} = ∥f∥_{L^{2}}. Also, inner products are preserved: ⟨Sf, Sg⟩_{H} = ⟨f, g⟩_{L^{2}}. Given a function f in L^{2}(0, 1) of the form f(x) = ∑_{n} c_{n}e_{n}(x), where only finitely many c_{n} are nonzero, we associate to it the function T_{φ}f(x) = ∑_{n} c_{n}φ_{n}(x), where φ_{n}(x) = φ(nx). It is clear that T_{φ}f ∈ L^{2}(0, 1), given the restriction on f. When the operator S is applied to T_{φ}f, we arrive at the identity

S(T_{φ}f)(s) = Sφ(s) Sf(s),  ℜs > \frac{1}{2}.  (5-1)

This has the interpretation that replacing the basis sequence {e_{n}}_{n} with {φ_{n}}_{n} corresponds to multiplication by Sφ on the S-transformed side.

5.1 The function 1/Sφ. The biorthogonal system. Let Sφ(s) = ∑_{n} a_{n}n^{−s}, as before. We formally write 1/Sφ(s) = ∑_{n} b_{n}n^{−s}. The coefficients b_{n} are given by b_{1} = 1,

b_{n} = ∑_{d_{1}d_{2}···d_{k}=n}(−1)^{k}a_{d_{1}}a_{d_{2}} ⋯ a_{d_{k}}, n > 1,

where the sum runs over all the finitely many possible product decompositions of n, where the various factors d_{j} are integers > 1. The series ∑_{n} b_{n}n^{−s} converges absolutely in some half-plane ℜs > σ_{0}, as is readily seen by considering the extremal situation where a_{n} ≤ 0 for all n > 1. In fact, any σ_{0} with ∑_{n>1} |a_{n}|n^{−σ_{0}} ≤ a_{1} = 1 will do. Multiplying Sφ with 1/Sφ, we read off from the coefficients that b_{1}a_{1} = 1 and

∑_{d:d|n} b_{d}a_{n/d} = 0,  n > 1.  (5-2)

Formally, we have 1/Sφ = Sψ, where ψ(x) = ∑_{n=1}^{∞} b_{n}e_{n}(x). It follows from (5-2) that the functions ψ_{n}(x) = ∑_{d:d|n} b̄_{n/d}e_{d}(x) (they are not the dilates of ψ) form a system which is biorthogonal to the original system {φ_{n}}_{n}, with φ_{n}(x) = φ(nx), that is, ⟨φ_{j}, ψ_{k}⟩ = δ_{j,k}, where δ_{j,k} is the Kronecker delta symbol, which is 1 when j = k, and 0 otherwise. It follows that the system {φ_{n}}_{n} is minimal, that is, each φ_{n} lies outside the closure of the linear span of the other vectors. Moreover, e_{n}(x) = ∑_{d:d|n} a_{n/d}ψ_{d}(x), and since these sums contain only a finite number of terms, it is clear that the biorthogonal system {ψ_{n}}_{n} is complete in L^{2}(0, 1). The biorthogonal system consists of dilations of a single function only in the trivial case φ(x) = e_{1}(x) = \sqrt{2} sin(πx).

5.2. Riesz bases. Recall that a basis {f_{n}}_{n} in a separable Hilbert space H is a Riesz basis if for some bounded invertible operator L on H, the sequence {Lf_{n}}_{n} is an orthonormal basis. An equivalent characterization is the following (see [21, pp. 30–37]).

Lemma 5.1. Let H be a separable Hilbert space. A system {f_{n}}_{n} of vectors from H is a Riesz basis in H if and only if

(a) every f ∈ H can be expanded as f = ∑_{n} c_{n}f_{n}.

(b) there are constants A and B, 0 < A ≤ B < +∞, such that

A( ∑_{n} |c_{n}|^{2})^{1/2} ≤ ∥ ∑_{n} c_{n}f_{n}∥_{L^{2}} ≤ B( ∑_{n} |c_{n}|^{2})^{1/2}  (5-3)

for every finite sequence of scalars c_{n}.

We now state the main result of this section.

Theorem 5.2. The system φ(x), φ(2x), φ(3x),… is a Riesz basis in L^{2}(0, 1) if and only if both Sφ and 1/Sφ belong to M.

Proof. As before, we write e_{n}(x) = \sqrt{2} sin(nπx), and let T_{φ} be the linear mapping which sends e_{n} to φ_{n}, for n = 1, 2, 3, . . .. Let 𝔉 denote the dense subspace of L^{2}(0, 1) consisting of functions f = ∑_{n} c_{n}e_{n}, where all but finitely many of the coefficients c_{n} equal 0. By Lemma 5.1, we have a Riesz basis if and only if the image of 𝔉 under T_{φ} is dense in L^{2}(0, 1), and

A ∥f∥_{L^{2}} ≤ ∥T_{φ}f∥_{L^{2}} ≤ B ∥f∥_{L^{2}},  f ∈ 𝔉,  (5-4)

holds for some constants A and B, 0 < A ≤ B < +∞. After an application of the transformation S, the Riesz basis condition becomes, if we use (5-1), that the image of S𝔉 under multiplication by Sφ should be dense in H, and that

A ∥Sf∥_{H} ≤ ∥Sφ Sf∥_{H} ≤ B ∥Sf∥_{H},  f ∈ 𝔉.  (5-5)

We first do the sufficiency part. If Sφ and 1/Sφ are both in M, then clearly, Sφ S𝔉 is dense, and (5-5) holds with A = ∥1/Sφ∥_{M}^{−1} and B = ∥Sφ∥_{M}.

We turn to the necessity part. By an argument involving Cauchy sequences, (5-4) extends to all f ∈ L^{2}(0, 1), so that Sφ is a multiplier on H. By (5-5) and the fact that Sφ S𝔉 is dense in H, 1/Sφ is a multiplier on H as well. The proof is complete. □

Example. In conjunction with Theorem 3.1, the theorem above provides a precise statement about the subtle dependence on the “closeness” of φ to e_{1}(x) = \sqrt{2} sin(πx). To illuminate the condition, consider

φ(x) = ∑_{n=1}^{∞} \frac{e_{n}(x)}{n^{τ}} = \sqrt{2} ∑_{n=1}^{∞} \frac{sin(nπx)}{n^{τ}},

for τ > \frac{1}{2}. Then

Sφ(s) = ∑_{n=1}^{∞} n^{−τ} n^{−s} = ζ(τ + s),

and

1/Sφ(s) = ∑_{n=1}^{∞} μ(n)n^{−τ} n^{−s},

where μ(n) is the Möbius function, which has μ(1) = 1, μ(n) = (−1)^{k} when n is the product of k different primes, and μ(n) = 0 if n is divisible by a square (other than 1). The function Sφ(s) is bounded in the half-plane ℜs > 0 if and only if τ > 1. Using Theorem 5.2 and the fact that multipliers are bounded analytic functions, we conclude that the system φ(x), φ(2x), … is a Riesz basis in L^{2}(0, 1) if and only if τ > 1.

The above example is covered by the following corollary to Theorem 5.2.

Corollary 5.3. If the coefficients a_{n} of Sφ ∈ H are totally multiplicative, the functions φ(x), φ(2x), φ(3x), … form a Riesz basis in L^{2}(0, 1) if and only if ∑_{p} |a_{p}| < +∞, where the sum runs over the primes.

Proof. Since the totally multiplicative coefficients come from a function in H, they must satisfy sup_{n} |a_{n}| < 1, and by the Euler product formula,

Sφ(s) = ∑_{n=1}^{∞} a_{n}n^{−s} = ∏_{p} (1 − a_{p}p^{−s})^{−1}.  (5-6)

By the remark following the proof of the multiplier theorem, M is isometrically isomorphic to H^{∞}(𝔻^{∞}). As we calculate the norm of 𝔔Sφ there, using the analog of (5-6), we obtain

∥Sφ∥_{M} = ∏_{p} (1 − |a_{p}|)^{−1},

so that in view of Theorem 5.2, ∑_{p} |a_{p}| < +∞ is certainly a necessary condition to have a Riesz basis. Incidentally, this is also sufficient, since

1/Sφ(s) = ∑_{n=1}^{∞} μ(n)a_{n}n^{−s} = ∏_{p} (1 − a_{p}p^{−s}). □

Note next the following consequence of Theorems 3.1 and 5.2; for the second statement, one should keep in mind the identification of M with H^{∞}(𝔻^{∞}) supplied by Section 3.

Corollary 5.4. If φ(x) = ∑_{n} a_{n}e_{n}(x), and ∑_{n>1} |a_{n}| < a_{1} = 1, then the collection {φ(nx)}_{n} (n = 1, 2, 3, …) forms a Riesz basis in L^{2}(0, 1). On the other hand, if a_{n} = 0 unless n is a prime or 1, the condition ∑_{n=2}^{∞} |a_{n}| < 1 is necessary to have a Riesz basis.

5.3. The completeness problem. The completeness problem is considerably more delicate than the Riesz basis problem. It does not seem likely that it has a solution as simple as the one given by Theorem 5.2. In fact, as we shall see, it is equivalent to the problem of describing the cyclic vectors in the space H^{2}(𝔻^{∞}), which is definitely quite hard. The relationship to the invariant subspaces in an infinite-dimensional setting was pointed out by the editors of Beurling’s collected works [1]. The level of difficulty of Beurling’s completeness problem is emphasized by the fact that a complete characterization of the cyclic vectors in H^{2}(𝔻^{N}) is known only when the (complex) dimension of the polydisk is N = 1.

A subspace I of H is said to be invariant if it is closed, and fg ∈ I whenever f ∈ I and g ∈ 𝔊. One shows, by suitably approximating functions in M with elements of 𝔊, that invariant subspaces are actually invariant under multiplication by elements of M. The invariant subspace generated by a function f ∈ H is denoted by [f]_{H}, and we say that f is cyclic if [f]_{H} = H. Similarly, a subspace J of H^{2}(𝔻^{∞}) is called invariant if it is closed, and invariant under multiplication by polynomials in the coordinate functions z_{1}, z_{2}, … on 𝔻^{∞}. The invariant subspaces are actually invariant under multiplication by H^{∞}(𝔻^{∞}). We write [f]_{H^{2}(𝔻^{∞})} for the invariant subspace generated by f ∈ H^{2}(𝔻^{∞}), and say that f is cyclic if [f]_{H^{2}(𝔻^{∞})} = H^{2}(𝔻^{∞}).

A moment’s reflection on the definition of T_{φ} reveals that φ_{1}, φ_{2}, φ_{3}, … form a complete system in L^{2}(0, 1) if and only if the image of 𝔉 under T_{φ} is dense in L^{2}(0, 1). After an application of the transformation S, using (5-1), the latter condition becomes the requirement that Sφ times the set 𝔊 = S𝔉 of finite Dirichlet series be dense in H, id est, that Sφ be cyclic in H. By the isometry between H and H^{2}(𝔻^{∞}) supplied by the operator 𝔔 back in Section 2, the following can be said.

Theorem 5.5. Let φ be as above. Then the following are equivalent:

(a) the system φ_{1}, φ_{2}, φ_{3}, … is complete in L^{2}(0, 1).

(b) the function Sφ is cyclic in H, that is, the subspace Sφ M is dense in H.

(c) the function 𝔔Sφ is cyclic in H^{2}(𝔻^{∞}).

A necessary condition for the completeness of the system {φ_{n}}_{n} mentioned by Beurling in his 1945 seminar (confer [1, p. 378]), is that the function Sφ(s) be zerofree in the half-plane ℜs > \frac{1}{2}. We shall prove something slightly stronger. For a sequence z = (z_{j})_{j} in 𝔻^{∞}, let ∥z∥_{l^{∞}} = sup{|z_{j}| : j = 1, 2, … } and ∥z∥_{l^{2}}^{2} = ∑_{j} |z_{j}|^{2}. We extend this notation to sets Ω ∈ 𝔻^{∞} by taking suprema over all the points in Ω.

Lemma 5.6. If the system φ_{1}, φ_{2}, … is complete in L^{2}(0, 1), and Ω ⊂ 𝔻^{∞} has ∥Ω∥_{l^{∞}} < 1 and ∥Ω∥_{l^{2}} < +∞, then

inf_{z∈Ω} |𝔔Sφ(z)| > 0.

Proof. In view of Theorem 5.5, we need to show that if g ∈ H^{2}(𝔻^{∞}) is cyclic, then |g| is bounded away from 0 on Ω. We argue by contradiction. So, suppose |g| is not bounded away from 0 on Ω; then there is a sequence of points z(k) in Ω such that g(z(k)) → 0 as k → +∞. When l^{2}(ℕ) is given the weak topology, its closed unit ball becomes a compact metric space (the Banach-Alaoglu theorem), so that by the Bolzano-Weierstraß theorem, the sequence {z(k)}_{k} possesses a cluster point z(∞), with ∥z(∞)∥_{l^{∞}} < 1 and ∥z(∞)∥_{l^{2}} < +∞. By the conditions on Ω, sup_{z∈Ω} ∏_{j}(1 − |z_{j}|^{2})^{−1} < +∞, so that point evaluations in Ω are uniformly bounded. It follows that as the subsequence of points z(k_{l}) converges to z(∞), the function values g(z(k_{l})) converge to g(z(∞)), so that g(z(∞)) = 0. We conclude that g cannot be cyclic, as it is annihilated by a bounded point evaluation. □

Remark. In particular, it follows that for the system φ_{1}, φ_{2}, … to be complete, a necessary condition is that

sup_{ℜs>σ_{0}} |Sφ(s)| > 0,

for any σ_{0} > \frac{1}{2}. This result is sharp, in the sense that for complete systems, inf_{ℜs> \frac{1}{2}} |Sφ(s)| = 0 can occur. An example is given after Corollary 5.8. A slightly more refined necessary condition for completeness can be obtained from Theorem 4.10: for all characters χ, the function (Sφ)_{χ}(s) = ∑_{n} a_{n}χ(n)n^{−s} must be outer in H_{∞}^{2}(ℂ_{\frac{1}{2}} ). Also, by a remark Helson makes in [16], it is necessary that (Sφ)_{χ} almost surely be outer in H_{i}^{2}(ℂ_{+}).

5.4. The Dirichlet-type space H_{d}. We introduce next a subspace of H which seems to play a natural role in the study of the completeness problem. To this end, let

d(n) = ∑_{d:d|n} 1

denote the number of divisors of a natural number n. As a function of n, the number of divisors is quite irregular, but it is well-known that d(n) = O(n^{δ}) holds for every δ > 0; however, the average order of d(n) is log n, confer [13, pp. 260–266]. We denote by H_{d} the collection of Dirichlet series f(s) = ∑_{n=1}^{∞} a_{n}n^{−s} for which

∑_{n=1}^{∞} |a_{n}|^{2}d(n) < ∞.

The set of those f in H_{d} for which a_{n} = 0 unless n is a power of a given prime p, coincides with the usual Dirichlet space on the unit disk, if we set z = p^{−s}.

Clearly, H_{d} ⊂ H. For the interesting class of series with totally multiplicative coefficients (a_{mn} = a_{m}a_{n}), we have the opposite inclusion. Indeed,

∑_{n} |a_{n}|^{2}d(n) = (∑_{n} |a_{n}|^{2})^{2}

in this favorable case. The identity can be verified by expanding both sums as Euler products. If we write n = ∏_{p} p^{ν_{p}}, where the product runs over the primes, and all but finitely many of the nonnegative integers ν_{p} are 0, we have

∑_{n} |a_{n}|^{2}d(n) = ∑_{n} |a_{2}^{ν_{2}}a_{3}^{ν_{3}} ⋯ a_{p}^{ν_{p}} ⋯ |^{2}(1 + ν_{2})(1 + ν_{3}) ⋯ (1 + ν_{p}) ⋯

= ∑_{ν_{2}=0}^{∞}(1 + ν_{2})|a_{2}|^{2ν_{2}} · ∑_{ν_{3}=0}^{∞}(1 + ν_{3})|a_{3}|^{2ν_{3}} ⋯ ∑_{ν_{p}=0}^{∞}(1 + ν_{p})|a_{p}|^{2ν_{p}} ⋯

= (1 − |a_{2}|^{2})^{−2} · (1 − |a_{3}|^{2})^{−2} ⋯ (1 − |a_{p}|^{2})^{−2} ⋯ = ∏_{p} (1 − |a_{p}|^{2})^{−2},

provided that each |a_{p}| < 1, and a_{1} = 1. Note that if some |a_{p}| ≥ 1, the sequence {a_{n}}_{n} is not even in l^{2}(ℕ). We have, by using a similar argument,

∑_{n=1}^{∞} |a_{n}|^{2} = ∏_{p} (1 − |a_{p}|^{2})^{−1},

where a_{1} = 1 and |a_{p}| < 1. The product converges if and only if

∑_{p} |a_{p}|^{2} < +∞, |a_{p}| < 1.

This is a necessary and sufficient criterion for a Dirichlet series with totally multiplicative coefficients to belong to the space H, and thus to the space H_{d}.

The reciprocal series of a Dirichlet series f(s) = ∑_{n} a_{n}n^{−s} with totally multiplicative coefficients and a_{1} = 1 is

\frac{1}{f(s)} = ∑_{n=1}^{∞} μ(n)a_{n}n^{−s},

where μ(n) is the Möbius function. A direct calculation yields

∑_{n=1}^{∞} |μ(n)a_{n}|^{2} = ∏_{p} (1 + |a_{p}|^{2}) ,  ∑_{n} |μ(n)a_{n}|^{2}d(n) = ∏_{p} (1 + 2|a_{p}|^{2})

for the expressions defining the norms in H and in H_{d}. These norms are finite if and only if ∑_{p} |a_{p}|^{2} < +∞. This criterion for 1/f(s) differs from the one for f(s) in one respect: the condition |a_{p}| < 1 is skipped.

5.5 Completeness, continued. We now use the space H_{d} to formulate a fairly general sufficient condition for completeness.

Theorem 5.7. The system {φ_{n}}_{n} is complete in L^{2}(0, 1) if one of the following conditions is fulfilled:

(a) Sφ and 1/Sφ are both in H_{d}.

(b) Sφ ∈ M and 1/Sφ ∈ H.

(c) Sφ ∈ H and 1/Sφ ∈ M.

Proof. We first look at parts (a) and (b). As before, we write 1/Sφ(s) = ∑_{n} b_{n}n^{−s}. Put R_{N}(s) = ∑_{n=1}^{N} b_{n}n^{−s}. Suppose we can show that ∥SφR_{N}∥_{H} has a bound which is independent of N. Then SφR_{N} tends to the constant function 1 weakly, as N → +∞. By standard functional analysis, 1 is in the norm closure of Sφ M, so that Sφ is cyclic in H. Under condition (b), it is clear that this boundedness condition is fulfilled, by the estimate ∥R_{N}∥_{H} ≤ ∥1/Sφ∥_{H}. Under condition (a), we apply the Cauchy-Schwarz inequality,

∥SφR_{N}∥_{H}^{2} = ∑_{k} | ∑_{{m,n: \atop mn=k,m≤N}} a_{n}b_{m}|^{2} ≤ ∑_{k} d(k) ∑_{m,n:mn=k} |a_{n}|^{2}|b_{m}|^{2}

≤ (∑_{n} |a_{n}|^{2}d(n)) (∑_{m} |b_{m}|^{2}d(m)) = ∥Sφ∥_{H}^{2}∥1/Sφ∥_{H}^{2},

where we used that d(mn) ≤ d(m)d(n). Part (a) follows. Under condition (c), we just multiply Sφ with the function 1/Sφ in M, to get that 1 ∈ [Sφ]_{H}, so that Sφ is cyclic. The proof is complete. □

Corollary 5.8. If the coefficients of Sφ ∈ H are totally multiplicative, the system {φ_{n}}_{n} is complete in L^{2}(0, 1).

Proof. According to the previous subsection, we have that both Sφ and 1/Sφ are in H_{d}, because the coefficients are totally multiplicative. The assertion now follows from part (a) of Theorem 5.7. □

Remark. It follows that in the context of the example considered after Theorem 5.2, we have completeness if and only if τ > \frac{1}{2}. In a way, this reproves a theorem of Helson [16]: the function Sφ is cyclic in H if τ > \frac{1}{2}.

Example. We use Theorem 5.7 to construct a complete system φ_{1}, φ_{2}, … with

inf_{ℜs> \frac{1}{2}} |Sφ(s)| = 0,

as promised in the remark following Lemma 5.6. Let b_{1} = 1, b_{2} = 0, and put b_{p} = p^{−1/2}(log log p)^{2/3} for prime indices p > 2, and extend the sequence multiplicatively, so that b_{mn} = b_{m}b_{n}. By Chebyshev’s theorem [19, Vol. 1, p. 25], the series

∑_{p} |b_{p}|^{2} = ∑_{p:p>2} \frac{1}{p (log log p)^{4/3}}

(summation over the primes) converges if and only if

∑_{n=3}^{∞} \frac{1}{n(log log n)^{4/3} log n}

converges. Since the last series converges by the integral test, we conclude from the results of the previous subsection that the functions Φ and 1/Φ are in H_{d}, where Φ(s) = ∑_{n} b_{n}n^{−s}. Let φ be given by Sφ = 1/Φ. The completeness now follows from Theorem 5.7. Again by Chebyshev’s theorem,

Φ(σ) ≥ ∑_{p:p>2} \frac{1}{p^{1/2+σ}(log log p)^{2/3}} ≥ A ∑_{n=3}^{∞} \frac{1}{n^{1/2+σ}(log log n)^{2/3} log n},

for some positive absolute constant A, whence Φ(σ) → +∞ as σ → \frac{1}{2}, since the right hand side series diverges for σ = \frac{1}{2}. It follows that Sφ(σ) → 0 as σ → \frac{1}{2}.

Theorem 5.9. Let Sφ(s) = 1 + ∑_{p} a_{p}p^{−s}, summing only over the primes. Then the system {φ_{n}}_{n} is complete in L^{2}(0, 1) if and only if ∑_{p} |a_{p}| ≤ 1.

Proof. To see the necessity, observe that if ∑_{p} |a_{p}| > 1, there exist a positive integer N and and a point z = (z_{1}, z_{2}, …) ∈ 𝔻^{∞} with z_{j} = 0 for j > N, such that (p_{j} is the j-th prime)

𝔔Sφ(z) = 1 + ∑_{j=1}^{∞} a_{p_{j}}z_{j} = 0.

By Lemma 5.6, we cannot have completeness.

To get the sufficiency, we introduce the auxiliary function g = 1 − Sφ, and note that ∥g∥_{M} ≤ 1. The function 1 − g^{n} = (1 − g)(1 + g + … + g^{n−1}) tends to 1 in the norm of H as n → +∞, and for each n, it is in Sφ M. It follows that Sφ is cyclic. □

6. Possible directions of further investigation

According to a theorem of Helson (Theorem 4.4), we have that for f ∈ H, with series expansion (2-1), the Dirichlet series

f_{χ}(s) = ∑_{n=1}^{∞} a_{n}χ(n)n^{−s}

converges in the half-plane ℜs > 0, for almost all characters χ. In terms of the coefficients, this amounts to having, almost surely in χ,

∑_{n=1}^{N} a_{n}χ(n) = O(N^{ε})  as N → +∞,  (6-1)

for each fixed ε > 0. In view of Theorem 4.5 and Lennart Carleson’s theorem on the almost everywhere convergence of Fourier series with l^{2} coefficients [6], which cover the probabilistic and deterministic extremes, the following conjecture seems reasonable: for almost all χ,

∑_{n=1}^{N} a_{n}χ(n) = O(1) as N → +∞.

The conjecture appears to be related to the question of how many of the stochastic variables χ(n) on a given interval N ≤ n < N + k are mutually independent. The latter translates into the following basic number-theoretic problem: What is the multiplicatively rational dimension of the interval N ≤ n < N + k? Here, the multiplicatively rational dimension of a set E of positive integers is the dimension of the ℚ-linear span of log E over the field ℚ. For instance, the multiplicatively rational dimension of the interval 1 ≤ n ≤ k equals the number of primes ≤ k.

By Corollary 4.7, almost every ζ_{χ}(s) converges to a zero-free analytic function in the half-plane ℜs > \frac{1}{2}. It will now be indicated how this result may be used to obtain information about particular characters. Given positive reals M and ε, let Ω(ε, M) be the set of all χ for which

| ∑_{n=1}^{N} χ(n)| ≤ M · N^{1/2+ε},

for all N = 1, 2, 3, . . .. By Corollary 4.8, the ρ-mass of Ω(ε, M) tends to 1 as M → +∞, for fixed ε. A point χ_{0} in Ξ is a mass point for Ω(ε, M) provided that each open neighborhood of χ_{0}, intersected with Ω(ε, M), has positive ρ-mass. If χ_{0} is a mass point for some Ω(ε, M), then ζ_{χ_{0}} extends analytically to a zero-free function on ℜs > \frac{1}{2} + ε. This is so because this function may be approximated by functions ζ_{χ}(s) which lack zeros on ℜs > \frac{1}{2}, in the topology of uniform convergence on compact subsets of ℜs > \frac{1}{2} + ε. This may be a way to handle non-principal Dirichlet characters, a topic to be discussed below.

As a matter of definition, a character is a multiplicative mapping χ : ℚ_{+} → 𝕋, and if we like, we may extend it to all of ℚ \ {0} by setting χ(−1) equal to 1 or −1. The characters that so extend continuously to the archimedian completion ℝ \ {0} are of the type χ(r) = |r|^{−it} for some real number t (if we fix χ(−1) = 1). There are also the non-archimedian completions ℚ_{p}, the p-adic number field, for a given fixed prime p. The Dirichlet characters associated with ℚ_{p} are the ones that extend continuously to ℚ_{p} \ {0}, and one checks that for some k ∈ ℕ, when restricted to the integers, they are periodic with period p^{k}, except on numbers divisible by p. Traditionally, one sets the value of the Dirichlet character χ equal to 0 on the integers divisible by p, but this is not our choice here. The principal Dirichlet characters take value 1 on all integers indivisible by p (there are several because we do not prescribe the value at p). The nonprincipal Dirichlet characters all have partial sums ∑_{n:n≤N} χ(n) that grow like O(log N) as N tends to infinity (with the traditional definition, these sums are actually bounded), so at first glance they seem like ideal candidates for mass points. In the final analysis, this may prove to be naïve.

Acknowledgement. The first author wishes to thank Paul Cohen (Stanford) for helpful conversations at an early stage of this work.

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Håkan Hedenmalm, Department of Mathematics, Uppsala University, Box 480, S– 75106 Uppsala, Sweden

E-mail address: haakan@math.uu.se

Peter Lindqvist, Department of Mathematical Sciences, The Norwegian Institute of Technology, N–7034 Trondheim, Norway

E-mail address: lqvist@imf.unit.no

Kristian Seip, Department of Mathematical Sciences, The Norwegian Institute of Technology, N–7034 Trondheim, Norway

E-mail address: seip@imf.unit.no