Proceedings of the Steklov Institute of Mathematics, Vol. 218, 1997, pp. 104–116. Translated from Trudy Matematicheskogo Instituta imeni V. A. Steklova, Vol. 218, 1997, pp. 109–121. Original Russian Text Copyright \begin{matrix}c \\ ◯ \end{matrix} 1997 by Garunkshtis, Laurinchicas. English Translation Copyright \begin{matrix}c \\ ◯ \end{matrix} by ���� �����/Interperiodica Publishing (Russia).

Limit Theorem with Weight for the Lerch Zeta Function in the Space of Analytic Functions

R. Garunkstis and A. Laurinchikas

Received February, 1997

Let s = σ + iτ be a complex variable and L(λ, α, s) denote the Lerch zeta function that, for σ > 1, is defined by the Dirichlet series

L(λ, α, s) = ∑_{m=0}^{∞} \frac{e^{2πiλm}}{(m + α)^{s}}

and outside of the plane σ > 1 by means of an analytic continuation. Here λ ∈ ℝ, 0 < α ≤ 1 are certain fixed parameters (ℝ is the set of real numbers). As usual, ℤ is the set of integers. The function L(λ, α, s) was introduced in the work [8], where it was found that for λ ∉ ℤ it was an entire function. To be more precise, it was shown that for all s

L(λ, α, s) = e^{−sπi}Γ(1 − s)\frac{1}{2πi}K(λ, α, s),

where, as usual, Γ(s) denotes Euler’s gamma-function, and

K(λ, α, s) = ∫_{L} \frac{e^{−λz}z^{s−1} dz}{1 − e^{2πiα−z}} .

The circuit L begins and terminates at infinity bypassing the point z = 0 along a circle. The points 2πi(α + k), k ∈ ℤ, do not lie on the circle L or inside it. The article [8] also contains a functional equation for the function L(λ, α, s).

When λ ∈ ℤ, the Lerch zeta-function obviously becomes Hurwitz zeta-function ζ(s, α), and when λ ∈ ℤ and α = 1, it become Riemannian zeta-function ζ(s). Thus the Lerch zeta-function is a generalization of the well-known classical functions.

The asymptotics of the moments of Lerch zeta-function was studied by Klusch [5, 6] and the asymptotics of the quantity

∫_{0}^{1} |L(λ, α, s) − α^{−s}|^{2} dα

was studied by Zhang [10] and Katsurada [4]. We became interested in the last work and began to investigate in [2] the values of the function L(λ, α, s) by statistical methods, where, in the case of λ ∉ ℤ, proved the limit theorem for L(λ, α, s) in the complex plane ℂ. The aim of this article is to obtain the limit theorem for L(λ, α, s) in the space of analytic functions.

Let D = {s ∈ ℂ : σ > 1/2} and let H(D) denote the space of functions analytic in D with the topology of uniform convergence on compacts. We denote by B(S) the class of Borel sets of the space S. Let T_{0} be a fixed positive number and let w(t) be a positive function of bounded variation on [T_{0}, ∞) such that its variation V_{a}^{b}w satisfies the inequality V_{a}^{b}w ≤ cw(a) with a certain constant c > 0 for all b > a ≥ T_{0}. We set

U = U(T, w) = ∫_{T_{0}}^{T} w(t) dt

and assume that

lim_{T →∞} U(T, w) = ∞.

Let us denote the probability measure

P_{T,w}(A) = \frac{1}{U} ∫_{T_{0}}^{T} w(τ)I_{\{τ:L(λ,α,s+iτ)∈A\}} dτ, A ∈ B(H(D)),

where I_{A} is the indicator of the set A. We assume, in what follows, that λ ∉ ℤ and α is a transcendental number.

Theorem. There exists on (H(D), B(H(D))) a probability measure P_{w} such that the measure P_{T,w} weakly converges to P_{w} as T → ∞.

A limit theorem of this kind for the Riemannian zeta-function was obtained in [7].

We preface the proof of the theorem with several lemmas. Note that the system of numbers {log(m + α), m = 0, 1, 2, … } is linearly independent over the field of rational numbers. Only in this statement do we use the transcendence of the number α.

We begin with proving the limit theorem for the Dirichlet polynomial

p_{n}(s, α) = ∑_{m=0}^{n} \frac{a_{m}}{(m + α)^{s}} ,

where a_{m} ∈ ℂ. When proving this theorem, we shall use the following properties of the weak convergence of measures.

Let S and S_{1} be two metric spaces and let h: S → S_{1} be a measurable function. Then every probability measure P on (S, B(S)) induces on (S_{1}, B(S_{1})) the unique probability measure P h^{−1} defined by the relation P h^{−1}(A) = P (h^{−1}A), A ∈ B(S_{1}).

Lemma 1. Let h: S → S_{1} be a continuous function and P_{n} and P be probability measures on (S, B(S)). If P_{n} weakly converges to P as n → ∞, then P_{n}h^{−1} also weakly converges to P h^{−1} as n → ∞.

Proof. The lemma is a corollary of Theorem 5.1 from [1].

Suppose that γ = {s ∈ ℂ : |s| = 1} and P is a probability measure on (γ^{r}, B(γ^{r})), where r is a positive integer. The Fourier transform g(k_{1}, …, k_{r}) of the measure P is defined by the relation

g(k_{1}, …, k_{r}) = ∫_{γ^{r}} x_{1}^{k_{1}} ⋯ x_{r}^{k_{r}} dP,

where k_{j} ∈ ℤ, x_{j} ∈ γ, j = 1, …, r.

Lemma 2. Let {P_{n}} be a sequence of probability measures on (γ^{r}, B(γ^{r})) and {g_{1}(k_{1}, …, k_{r})} be the sequence of the corresponding Fourier transforms. We assume that for every collection of integers (k_{1}, …, k_{r}) there exists a limit

g(k_{1}, …, k_{r}) = lim_{n→∞} g_{n}(k_{1}, …, k_{r}).

Then there exists on (γ^{r}, B(γ^{r})) a probability measure P such that P_{n} weakly converges to P as n → ∞. In this case, g(k_{1}, …, k_{r}) is a Fourier transform of the measure P.

Proof. The lemma is a special case of the continuity theorems for compact Abelian groups, see, e.g., [11].

Let us define on (H(D), B(H(D))) the probability measure

P_{T,n,w}(A) = \frac{1}{U} ∫_{T_{0}}^{T} w(τ)I_{\{τ:p_{n}(s+iτ,α)∈A\}} dτ.

Lemma 3. There exists on (H(D), B(H(D))) a probability measure P_{n,w} such that the measure P_{T,n,w} weakly converges to P_{n,w} as T → ∞.

Proof. Let

Ω_{n} = ∏_{m=0}^{n} γ_{m},

where γ_{m} = γ for all m = 0, 1, …, n. We define the function h: Ω_{n} → H(D) by the relation

h(x_{0}, …, x_{n}) = ∑_{m=0}^{n} \frac{a_{m}}{(m + α)^{s}x_{m}} , (x_{0}, …, x_{n}) ∈ Ω_{n}.

Then the function h is continuous, and we have the relation

p_{n}(s + iτ, α) = h(α^{iτ} , (1 + α)^{iτ} , …, (n + α)^{iτ} ).  (1)

Let us define the probability measure

Q_{T,w}(A) = \frac{1}{U} ∫_{T_{0}}^{T} w(τ)I_{\{τ:(α^{iτ} ,(1+α)^{iτ} ,...,(n+α)^{iτ})∈A\}} dτ, A ∈ B(Ω_{n}),

and denote by g_{T} (k_{0}, …, k_{n}), k_{j} ∈ ℤ, j = 0, …, n, its Fourier transform. Then we have

g_{T} (k_{0}, …, k_{n}) = ∫_{Ω_{n}} x_{0}^{k_{0}} ⋯ x_{n}^{k_{n}} dQ_{T,w} = \frac{1}{U} ∫_{T_{0}}^{T} w(τ) ∏_{m=0}^{n}(m + α)^{iτk_{m}} dτ.

Integrating by parts and using the properties of the function w(τ) and the linear independence of the system {log(m + α), m = 0, 1, … }, we find that

∫_{T_{0}}^{T} w(τ) ∏_{m=0}^{n}(m + α)^{iτk_{m}}dτ = (i ∑_{m=0}^{n} k_{m} log(m + α))^{−1} ∫_{T_{0}}^{T} w(τ) d exp{iτ ∑_{m=0}^{n} k_{m} log(m + α)}

= B|∑_{m=0}^{n} k_{m} log(m + α)|^{−1} − (i ∑_{m=0}^{n} k_{m} log(m + α))^{−1} ∫_{T_{0}}^{T} exp{iτ ∑_{m=0}^{n} k_{m} log(m + α)} dw(τ)

= B|∑_{m=0}^{n} k_{m} log(m + α)|^{−1}

for (k_{0}, …, k_{n}) ≠ (0, …, 0). Here and in the sequel B_{η} denotes a quantity (not always the same) bounded modulo by a constant dependent on the parameter η. Thus

g_{T} (k_{0}, …, k_{n}) = \{ \begin{matrix}1, & (k_{0}, …, k_{n}) = (0, …, 0) \\ B(U|∑_{m=0}^{n} k_{m} log(m + α)|)^{−1}, & (k_{0}, …, k_{n}) ≠ (0, …, 0). \end{matrix}

Consequently,

lim_{T →∞} g_{T} (k_{0}, …, k_{n}) = \{ \begin{matrix}1, & (k_{0}, …, k_{n}) = (0, …, 0) \\ 0, & (k_{0}, …, k_{n}) ≠ (0, …, 0). \end{matrix}

By virtue of Lemma 2, we find that the measure Q_{T,w} weakly converges to a certain measure m_{n} on (Ω_{n}, B(Ω_{n})) as T → ∞ (the measure m_{n} is the Haar measure). Therefore we find from Lemma 1, the continuity of the function h, and relation (1) that the measure P_{T,n,w} weakly converges to m_{n}h^{−1} as T → ∞.

We can assume, without loss of generality, that 0 < λ < 1. In order to approximate the function L(λ, α, s) in the mean by the absolutely convergent Dirichlet series, we shall need the following two lemmas. The first of them is an analog of the approximation of the Riemannian zeta-function by a finite sum, see, for instance, [3].

Lemma 4. Let 0 < σ_{0} ≤ σ ≤ 2 and |t| ≤ πλx. Then

L(λ, α, s) = ∑_{0≤m≤x} \frac{e^{2πiλm}}{(m + α)^{s}} + B_{σ_{0}}x^{−σ}.

The proof of the lemma for the case 0 < σ_{0} ≤ σ ≤ 1, 0 ≤ t ≤ πλx is given in [2]. However, it is easy to see that it remains valid in the present formulation.

Lemma 5. Let σ > 1/2 be a fixed number. Then, for every τ ∈ ℝ, we have

∫_{T_{0}+τ}^{T +τ} w(t − τ)|L(λ, α, σ + it)|^{2} dt = B_{λ,α}U(1 + |τ|)^{2}.

Proof. It is obvious that for σ > 2 the integral of the lemma is estimated as BU. Therefore we shall only consider the case σ ≤ 2. Setting x = (t + |τ|)λ^{−1} in Lemma 4, we obtain

∫_{T_{0}+τ}^{T +τ} w(t − τ)|L(λ, α, σ + it)|^{2} dt = B ∫_{T_{0}}^{T} w(t)| ∑_{m≤(t+|τ|)λ^{−1}} \frac{e^{2πiλm}}{(m + α)^{σ+it+iτ}} |^{2} dt + B_{λ}U. (2)

Let max(m, k) = (T_{1} + |τ|)λ^{−1}, where T_{1} = T_{1}(m, k). Then

∫_{T_{0}}^{T} w(t)| ∑_{m≤(t+|τ|)λ^{−1}} \frac{e^{2πiλm}}{(m + α)^{σ+it+iτ}} |^{2} dt

= ∑ ∑_{\begin{matrix}(T_{0}+|τ|)λ^{−1}<m≤(T +|τ|)λ^{−1} \\ (T_{0}+|τ|)λ^{−1}<k≤(T +|τ|)λ^{−1} \end{matrix}} \frac{e^{2πiλ(m−k)}}{(m + α)^{σ+iτ} (k + α)^{σ−iτ}} ∫_{T_{1}}^{T} w(t)(\frac{k + α}{m + α})^{it} dt + B_{λ}U(1 + |τ|)

= ∑_{(T_{0}+|τ|)λ^{−1}<m≤(T +|τ|)λ^{−1}} \frac{1}{(m + α)^{2σ}} ∫_{T_{1}}^{T} w(t) dt

+ B ∑ ∑_{(T_{0}+|τ|)λ^{−1}<m<k≤(T +|τ|)λ^{−1}} \frac{w(k − |τ|)}{(m + α)^{σ}(k + α)^{σ} log \frac{k+α}{m+α}} + B_{λ}U(1 + |τ|)

= B_{λ}U(1 + |τ|) + B ∑ ∑_{T_{0}λ^{−1}<m<k≤T λ^{−1}} \frac{w(k)}{(m + |τ|)^{σ}(k + |τ|)^{σ} log \frac{k+|τ|+α}{m+|τ|+α}} .  (3)

First suppose that m + |τ| + α < (k + |τ| + α)/2. Then

log \frac{k + |τ| + α}{m + |τ| + α} > log 2,

and, consequently,

∑ ∑_{\begin{matrix}T_{0}λ^{−1}<m<k≤T λ^{−1} \\ m+|τ|+α<(k+|τ|+α)/2 \end{matrix}} \frac{w(k)}{(m + |τ|)^{σ}(k + |τ|)^{σ} log \frac{k+|τ|+α}{m+|τ|+α}} = B ∑_{T_{0}λ^{−1}<m<k≤T λ^{−1}} \frac{w(k)}{m^{σ}k^{σ}}
= B ∑_{T_{0}λ^{−1}<k≤T λ^{−1}} w(k) = B_{λ}U,  (4)

since

∑_{T_{0}λ^{−1}<k≤T λ^{−1}} w(k) = ∫_{T_{0}λ^{−1}}^{T λ^{−1}}w(ω) d[u] = B_{λ} + w(T λ^{−1})[T λ^{−1}] − ∫_{T_{0}λ^{−1}}^{T λ^{−1}}(u − {u}) dw(u)
= B_{λ} + w(T λ^{−1})([T λ^{−1}] − T λ^{−1}) + ∫_{T_{0}λ^{−1}}^{T λ^{−1}}w(u) du = B_{λ}U.  (5)

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

LIMIT THEOREM WITH WEIGHT FOR LERCH ZETA FUNCTION 109

If m + |τ| + α ≥ (k + |τ| + α)/2, then we set m = k − r, where 1 ≤ r ≤ (k + |τ| + α)/2. Then

log \frac{k + |τ| + α}{m + |τ| + α} = − log \frac{k − r + |τ| + α}{k + |τ| + α} = − log(1 − \frac{r}{k + |τ| + α}) > \frac{r}{k + |τ| + α}.

Concequently, by virtue of (5), we have

∑ ∑_{\begin{matrix}T_{0}λ^{−1}<m<k≤T λ^{−1} \\ m+|τ|+α≥(k+|τ|+α)/2 \end{matrix}} \frac{w(k)}{(m + |τ|)^{σ}(k + |τ|)^{σ} log \frac{k+|τ|+α}{m+|τ|+α}}
= B ∑_{T_{0}λ^{−1}<k≤T λ^{−1}} ∑_{r≤(k+|τ|+α)/2} \frac{w(k)(k + |τ|)}{(k + |τ| − r)^{σ}k^{σ}r}
= B ∑_{T_{0}λ^{−1}<k≤T λ^{−1}} \frac{w(k)}{k^{2σ−1}} ∑_{r≤(k+|τ|+α)/2} \frac{1}{r} + B(1 + |τ|) ∑_{T_{0}λ^{−1}<k≤T λ^{−1}} \frac{w(k)}{k^{2σ}} ∑_{r≤(k+|τ|+α)/2} \frac{1}{r}
= B(1 + |τ|) ∑_{T_{0}λ^{−1}<k≤T λ^{−1}} w(k)\frac{log k}{k^{2σ−1}} + B(1 + |τ|)^{2} ∑_{T_{0}λ^{−1}<k≤T λ^{−1}} w(k)\frac{log k}{k^{2σ}}
= B_{λ}(1 + |τ|)^{2}U.

From this and from (2)–(4) we get the statement of the lemma.

Suppose now that σ_{1} > 1/2. In the strip σ ∈ [−σ_{1}, σ_{1}] we shall define the function

l_{n}(α, s) = \frac{s}{σ_{1}} Γ(\frac{s}{σ_{1}} )(n + α)^{s}, n = 1, 2, ….

In the half-plane σ > 1/2 we set

L_{n}(λ, α, s) = \frac{1}{2πi} ∫_{σ_{1}−i∞}^{σ_{1}+i∞}L(λ, α, s + z)l_{n}(α, z) \frac{dz}{z} .  (6)

It follows from Lemma 4 that

L(λ, α, s) = B_{λ,α}(1 + |t|).

for σ ≥ 1/2. Therefore, by virtue of the estimate

Γ(σ + it) = Be^{−c_{1}|t|}, |t| ≥ c_{2},

which is uniform with respect to σ, |σ| ≤ c_{3}, the integral in (6) exists.

Lemma 6. The relation

L_{n}(λ, α, s) = ∑_{m=0}^{∞} \frac{e^{2πiλm}}{(m + α)^{s}} exp\{−(\frac{m + α}{n + α})^{σ_{1}}\}

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

110 GARUNKSHTIS, LAURINCHICAS

holds true, and the series converges absolutely for σ > 1/2.

Proof. It is obvious that σ + σ_{1} > 1 for σ > 1/2. Therefore the function L(λ, α, s + z) on the straight line Re z = σ_{1} is represented by the absolutely convergent Dirichlet series

L(λ, α, s + z) = ∑_{m=0}^{∞} \frac{e^{2πiλm}}{(m + α)^{s+z}} .

We set

a_{n}(α, m) = \frac{1}{2πi} ∫_{σ_{1}−i∞}^{σ_{1}+i∞}\frac{l_{m}(α, s) ds}{s(m + α)^{s}}

and consider the series

∑_{m=0}^{∞} \frac{a_{n}(α, m)e^{2πiλm}}{(m + α)^{s}} .  (7)

It is clear that

a_{n}(α, m) = B(m + α)^{−σ_{1}} ∫_{−∞}^{∞} |l_{n}(α, σ_{1} + it)| dt = B(m + α)^{−σ_{1}}.

This estimate gives the absolute convergence of series (7) for σ > 1/2. Consequently, changing the order of integration and summation in (6), we obtain

L_{n}(λ, α, s) = ∑_{m=0}^{∞} \frac{a_{n}(α, m)e^{2πiλm}}{(m + α)^{s}} .  (8)

By virtue of the well-known property of the gamma-function

\frac{1}{2πi} ∫_{b−i∞}^{b+i∞}Γ(s)a^{−s} ds = e^{−a},

which is valid for all positive a and b, we find that

a_{n}(α, m) = \frac{1}{2πi} ∫_{σ_{1}−i∞}^{σ_{1}+i∞}Γ(\frac{s}{σ_{1}} )(\frac{m + α}{n + α})^{−s} d(\frac{s}{σ_{1}} ) = exp\{−(\frac{m + α}{n + α})^{σ_{1}}\}.

The last relation, together with (8), proves the lemma.

We shall prove now that the function L(λ, α, s) in D can be approximated by the series L_{n}(λ, α, s).

Lemma 7. Let K be a compact set of the half-plane D. Then

lim_{n→∞} \overline{lim}_{T →∞} \frac{1}{U} ∫_{T_{0}}^{T} w(τ) sup_{s∈K} |L(λ, α, s + iτ) − L_{n}(λ, α, s + iτ)| dτ = 0.

Proof. Let us shift the integration line in relation (6). The integrand function has a simple pole at the point z = 0. Let σ ∈ [1/2+η, A], η > 0, when s ∈ K. We set σ_{2} = 1/2+η/2. Then, in accordance with the residue theorem, we obtain

L_{n}(λ, α, s) = L(λ, α, s) + \frac{1}{2πi} ∫_{σ_{2}−σ−i∞}^{σ_{2}−σ+i∞}L(λ, α, s + z)l_{n}(α, z) \frac{dz}{z} .  (9)

Let L be a simple closed circuit which lies in D and encircles the set K and δ be the distance from L to K. Then, by virtue of Cauchy’s integral formula, we obtain

sup_{s∈K} |L(λ, α, s + iτ) − L_{n}(λ, α, s + iτ)| ≤ \frac{1}{2πδ} ∫_{L} |L(λ, α, z + iτ) − L_{n}(λ, α, z + iτ)| |dz|.

Let |L| be the length of the circuit L. Then it follows that

\frac{1}{U} ∫_{T_{0}}^{T} w(τ) sup_{s∈K} |L(λ, α, s + iτ) − L_{n}(λ, α, s + iτ)| dτ

= \frac{B}{Uδ} ∫_{L} |dz| ∫_{T_{0}+Im z}^{T +Im z}w(τ − Im z)|L(λ, α, Re z + iτ) − L_{n}(λ, α, Re z + iτ)| dτ

= \frac{B|L|}{Uδ} sup_{σ+iu∈L} ∫_{T_{0}+u}^{T +u}w(t − u)|L(λ, α, σ + it) − L_{n}(λ, α, σ + it)| dt.  (10)

We can choose the circuit L such that the inequalities inf{σ : s ∈ L} ≥ 1/2 + 3η/4, δ ≥ η/4 are satisfied. From relation (9) we have

L(λ, α, σ + it) − L_{n}(λ, α, σ + it) = B ∫_{−∞}^{∞} |L(λ, α, σ_{2} + it + iτ)| × |l_{n}(α, σ_{2} − σ + iτ)| dτ.

Thus

\frac{1}{U} ∫_{T_{0}+u}^{T +u}w(t − u)|L(λ, α, σ + it) − L_{n}(λ, α, σ + it)| dt

= B ∫_{−∞}^{∞} |l_{n}(α, σ_{2} − σ + iτ)|\frac{1}{U} ∫_{T_{0}+u+τ}^{T +u+τ} w(t − u − τ)|L(λ, α, σ_{2} + it)| dt dτ.  (11)

Using the Cauchy inequality, we obtain

∫_{T_{0}+u+τ}^{T +u+τ} w(t−u−τ)|L(λ, α, σ_{2}+it)| dt ≤ (∫_{T_{0}}^{T} w(t) dt ∫_{T_{0}+u+τ}^{T +u+τ} w(t−u−τ)|L(λ, α, σ_{2}+it)|^{2} dt)^{1/2}. (12)

Since the quantity u is bounded, we have, by virtue of Lemma 5,

∫_{T_{0}+u+τ}^{T +u+τ} w(t − u − τ)|L(λ, α, σ_{2} + it)|^{2} dt = BU(1 + |τ|)^{2}.

From this and from (11) and (12) we find that the left-hand side of function (10) does not exceed

B sup_{σ+iu∈L} ∫_{−∞}^{∞} |l_{n}(α, σ_{2} − σ + iτ)|(1 + |τ|) dτ = B sup_{σ∈[−A,−η/4]} ∫_{−∞}^{∞} |l_{n}(α, σ_{2} − σ + iτ)|(1 + |τ|) dτ.

However, it follows from the definition of l_{n}(α, s) that

lim_{n→∞} sup_{σ∈[−A,−η/4]} ∫_{−∞}^{∞} |l_{n}(α, σ_{2} − σ + iτ)|(1 + |τ|) dτ = 0.

We have proved the lemma.

Now we shall prove the limit theorem for the function L_{n}(λ, α, s). We shall use the following well-known results.

Lemma 8. If the family of functions regular in the domain G is uniformly bounded inside G, then it is compact in G.

Proof. The lemma is known as compactness principle, its proof can be found, for instance, in [9].

Let Y_{n}, X_{1n}, X_{2n}, … be (S, ϱ)-valued random elements defined on the same probability space (Ω, F, �). We assume that S is a separable space.

Lemma 9. Let X_{kn} →^{D} X_{k}, as n → ∞, for every k and let X_{k} →^{D} X as k → ∞. In addition, suppose that

lim_{k→∞} \overline{lim}_{n→∞} �\{ϱ(X_{kn}, Y_{n}) ≥ ε\} = 0

for every ε > 0. Then Y_{n} →^{D} X as n → ∞.

The proof is given in [1].

Let us define on (H(D), B(H(D))) the probability measure by means of the function L_{n}(λ, α, s),

Q_{T,n,w}(A) = \frac{1}{U} ∫_{T_{0}}^{T} w(τ)I_{\{τ:L_{n}(λ,α,s+iτ)∈A\}} dτ,

and consider its weak convergence as T → ∞.

Lemma 10. There exists on (H(D), B(H(D))) a probability measure Q_{n,w} such that the measure Q_{T,n,w} weakly converges to Q_{n,w} as T → ∞.

Proof. Also suppose that

L_{n,N}(λ, α, s) = ∑_{m≤N} \frac{e^{2πiλm}}{(m + α)^{s}} exp\{−(\frac{m + α}{n + α})^{σ_{1}}\}

and

Q_{T,n,N,w}(A) = \frac{1}{U} ∫_{T_{0}}^{T} w(τ)I_{\{τ:L_{n,N} (λ,α,s+iτ)∈A\}} dτ, A ∈ B(H(D)).

Then it follows from Lemma 3 that as T → ∞ the measure Q_{T,n,N,w} weakly converges to a certain measure Q_{n,N,w} on (H(D), B(H(D))). We shall show that the family {Q_{n,N,w}} is dense for every fixed n.

Let η be a random variable on (Ω̃, F, �) such that

�\{η ∈ A\} = \frac{1}{U} ∫_{T_{0}}^{T} w(t)I_{A} dt, A ∈ B(ℝ).

We define the H(D)-valued random element X_{T,n,N} by the relation

X_{T,n,N}(s) = L_{n,N}(λ, α, s + iη).

Then we find that

�\{X_{T,n,N}(s) ∈ A\} = \frac{1}{U} ∫_{T_{0}}^{T} w(τ)I_{\{τ:L_{n,N} (λ,α,s+iτ)∈A\}} dτ, A ∈ B(H(D)).

Consequently,

X_{T,n,N} ⟶^{D}_{T →∞} X_{n,N},  (13)

where X_{n,N} is a H(D)-valued random element with distribution Q_{n,N,w}.

Let us introduce on H(D) the metric ϱ as follows:

ϱ(f_{1}, f_{2}) = ∑_{n=1}^{∞} 2^{−n} \frac{ϱ_{n}(f_{1}, f_{2})}{1 + ϱ_{n}(f_{1}, f_{2})}, f_{1}, f_{2} ∈ H(D),

where

ϱ_{n}(f_{1}, f_{2}) = sup_{s∈K_{n}} |f_{1}(s) − f_{2}(s)|

and {K_{n}} is a sequence of compact subsets from D such that

D = ⋃_{n=1}^{∞} K_{n}, K_{n} ⊂ K_{n+1}, n = 1, 2, …,

and if K is a compact subset from D, then K ⊆ K_{n} for some n. It is easy to see that the metric that we introduced induces the topology of the space H(D).

The Chebyshev inequality yields

�\{ sup_{s∈K_{l}} |X_{T,n,N}(s)| > M_{l}\} ≤ \frac{1}{M_{l}U} ∫_{T_{0}}^{T} w(τ) sup_{s∈K_{l}} |L_{n,N}(λ, α, s + iτ)| dτ.  (14)

It follows from the absolute convergence of the series for L_{n}(λ, α, s) in the half-plane D that

sup_{N≥1} \overline{lim}_{T →∞} \frac{1}{U} ∫_{T_{0}}^{T} w(τ) sup_{s∈K_{l}} |L_{n,N}(λ, α, s + iτ)| dτ ≤ A_{l} < ∞.

Consequently, setting M_{l} = A_{l}2^{l}/ε, we find from (14) for every ε > 0 that

\overline{lim}_{T →∞} �\{ sup_{s∈K_{l}} |X_{T,n,N}(s)| > M_{l}\} ≤ \frac{ε}{2^{l}}  (15)

for all positive integers l. The function h: H(D) → ℝ, defined by the relation

h(f) = sup_{s∈K_{l}} |f(s)|, f ∈ H(D),

is continuous. Thus Lemma 1 and relation (13) imply

sup_{s∈K_{l}} |X_{T,n,N}(s)| ⟶^{D}_{T →∞} sup_{s∈K_{l}} |X_{n,N}(s)|.

From this and from (15) we find that

�\{ sup_{s∈K_{l}} |X_{n,N}(s)| > M_{l}\} ≤ \frac{ε}{2^{l}}  (16)

for all positive integers l. We set

H_{ε} = {f ∈ H(D) : sup_{s∈K_{l}} |f(s)| ≤ M_{l}, l ≥ 1}.

Then, from the properties of the sets K_{l} and from Lemma 8 we find that H_{ε} is a compact set of the space H(D), and, by virtue of (16), it follows that

�{X_{n,N}(s) ∈ H_{ε}} ≥ 1 − ε

for all N ≥ 1, or

Q_{n,N,w}(H_{ε}) ≥ 1 − ε

for all N ≥ 1. The latter means that the family of probability measures {Q_{n,N,w}} is dense. From this we find, in accordance with the Prokhorov theorem, that {Q_{n,N,w}} is relatively compact.

It is clear that L_{n,N}(λ, α, s) converges to L_{n}(λ, α, s) uniformly as N → ∞ when σ ≥ \frac{1}{2} + δ for any δ > 0. Consequently,

lim_{N→∞} \overline{lim}_{T →∞} \frac{1}{U} ∫_{T_{0}}^{T} w(τ)I_{\{τ:ϱ(L_{n,N} (λ,α,s+iτ),L_{n}(λ,α,s+iτ))≥ε\}} dτ

≤ lim_{N→∞} \overline{lim}_{T →∞} \frac{1}{εU} ∫_{T_{0}}^{T} w(τ)ϱ(L_{n,N}(λ, α, s + iτ), L_{n}(λ, α, s + iτ)) dτ = 0.

Suppose, in addition, that

X_{T,n}(s) = L_{n}(λ, α, s + iη).

Then, again using absolute convergence of the series for L_{n}(λ, α, s) and Chebyshev’s inequality, we find that

lim_{N→∞} \overline{lim}_{T →∞} �\{ϱ(X_{T,n,N}(s), X_{T,n}(s)) ≥ ε\} = 0.  (17)

It follows from the relative compactness of the family {Q_{n,N,w}} that there exists a subsequence {Q_{n,N_{1},w}} that weakly converges as N_{1} → ∞, say, to Q_{n,w}. Then we have

X_{n,N_{1}} ⟶^{D}_{N_{1}→∞} Q_{n,w}.  (18)

The space H(D) is separable, and therefore it follows from (13), (17), (18), and from Lemma 9 that

X_{T,n} ⟶^{D}_{T →∞} Q_{n,w}.  (19)

The latter means that as T → ∞, the measure Q_{T,n,w} weakly converges to the measure Q_{n,w}.

Proof of the Theorem. We shall show that the family of probability measures {Q_{n,w}, n ≥ 1} is dense. From (19) we have

X_{T,n} ⟶^{D}_{T →∞} X_{n},  (20)

where X_{n} is a H(D)-valued random element with distribution Q_{n,w}. Reasoning as in the proof of Lemma 9 and preserving its notation, we find that

�{X_{n}(s) ∈ H_{ε}} ≥ 1 − ε

for all n ≥ 1, i.e., we find that the family {Q_{n,w}} is dense.

We define a subsequence {Q_{n_{1},w}} from {Q_{n,w}} such that as n_{1} → ∞, Q_{n_{1},w} weakly converges, say, to Q_{w}. Then

X_{n_{1}} ⟶^{D}_{n_{1}→∞} Q_{w}.  (21)

Using Lemma 7 and Chebyshev’s inequality, we derive

lim_{n→∞} \overline{lim}_{T →∞} \frac{1}{U} ∫_{T_{0}}^{T} w(τ)I_{\{τ:ϱ(L(λ,α,s+iτ),L_{n}(λ,α,s+iτ))≥ε\}} dτ

≤ lim_{n→∞} \overline{lim}_{T →∞} \frac{1}{εU} ∫_{T_{0}}^{T} w(τ)ϱ(L(λ, α, s + iτ), L_{n}(λ, α, s + iτ)) dτ = 0.

Setting

X_{T} (s) = L(λ, α, s + iη),

we find that

lim_{n→∞} \overline{lim}_{T →∞} �\{ϱ(X_{T,n}(s), X_{T} (s)) ≥ ε\} = 0.

The last relation, (20), (21), and Lemma 9 yield

X_{T} ⟶^{D}_{T →∞} Q_{w},

and this proves the theorem.

ACKNOWLEDGMENTS

This work was supported by the Lithuanian Foundation for Research and Studios.

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Translated by I. Aleksandrova