MATHEMATICS

B. S. BRONSTEIN

ON THE DISTRIBUTION OF SINGULARITIES OF ONE CLASS OF FUNCTIONS

(Presented by Academician I. M. Vinogradov on 23 V 1960)

The theorem on the distribution of singularities of one class of Dirichlet series of the form

[
f(s)=\sum_{n=1}^{\infty} a_n \exp(-\lambda_n s), \qquad
0\leq \lambda_1<\lambda_2<\cdots,
\tag{1}
]

with the condition on the exponents

[
\lim(\lambda_{n+1}-\lambda_n)\geq h>0,
\tag{2}
]

proved by Agmon ((^3)), is generalized in the present paper to a broader class of functions. In addition, a certain refinement of this theorem is given.

Definition. We shall say that a function (f(s)) belongs to the class (B(r,h)) if (f(s)) is representable for (\sigma>0) ((s=\sigma+it)) as the limit (uniformly convergent in any bounded closed domain belonging to the half-plane (\sigma>0)) of a sequence of “Dirichlet polynomials”:

[
D_k(s)=\sum_{n=1}^{n_k}\sum_{j=0}^{\alpha_n-1} a_{n,k,j}\, s^j \exp(-\gamma_n s),
\tag{3}
]

where the sequence of exponents ({\lambda_n}), in which each (\gamma_n) occurs (\alpha_n) times, satisfies the condition

[
\lim(\lambda_{n+r}-\lambda_n)\geq h>0.
\tag{4}
]

We shall denote by ([e^{-xs},\delta_0,\delta_1,\ldots,\delta_n]) the finite divided difference of the function (e^{-xs}) at the points (\delta_0,\delta_1,\ldots,\delta_n). From the results of V. Bernstein ((^4)) and A. F. Leont’ev ((^8)) it follows that a function (f(s)) of class (B(r,h)) is representable for (\sigma>0) by an absolutely convergent series

[
f(s)=\sum_{n=1}^{\infty} A_n(s),
\tag{5}
]

where

[
A_k(s)=\sum_{\nu=0}^{p_k}\alpha_{k,\nu}
[e^{-xs},\lambda_{m_k},\lambda_{m_k+1},\ldots,\lambda_{m_k+\nu}],
\tag{6}
]

and the following properties hold: (\alpha)) (p_k0) such that (\lambda_{m_k+1}-\lambda_{m_k+p_k}\geq h_1) for all (k); (\delta)) there is a number (H) such that for all (k), (\lambda_{m_k+1}-\lambda_{m_k}0).

A function (f(s)) of class (B(r,h)) we shall also call a series of class (B(r,h)), meaning the series (5).

Let (\lim_{k\to\infty}\alpha_k=\infty). From properties a)—e) and geometric considerations it follows that there exists the least nonconvex (downward) envelope (C(x)) of the points (P_k(\lambda_{m_k},\log\alpha_k)) (the possibility is not excluded that (\log\alpha_k=-\infty)), and (C(x)) is a broken line with an infinite number of vertices at points of the form (P_{k_i}(\lambda_{m_{k_i}},\log\alpha_{k_i})). Denote (q_k=\exp C(\lambda_{m_k})). Then for all (k), (\alpha_k\leq q_k), and for an infinite set of indices (k_i),

[
q_{k_i}=\alpha_{k_i}.
\tag{7}
]

Theorem 1. Let (D) be a simply connected domain in which the function (f(s)) of class (B(r,h)) is regular.

Then the family of functions ({f_k(s)}), where

[
f_k(s)=\left[f(s)-\sum_{n=1}^{k-1} A_n(s)\right]q_k^{-1}\exp(\lambda_{m_k}s),
\tag{8}
]

is uniformly bounded in every closed bounded domain belonging to (D).

The proof scheme is the same as for Theorem 1 in ((^1)), using a)—e) and the easily derived estimates:

[
\left|[e^{-xs},\delta_0,\delta_1,\ldots,\delta_n]\right|
\leq |s|^n\exp(-\delta_0\sigma)\quad \text{for } \sigma>0;
\tag{9}
]

[
\left|[e^{-xs},\delta_0,\delta_1,\ldots,\delta_n]\right|
\leq |s|^n\exp(-\delta_n\sigma)\quad \text{for } \sigma<0.
\tag{10}
]

Theorem 2. Suppose that, under the conditions of Theorem 1, the imaginary axis is not a natural boundary of (f(s)).

Then every limit function (g(s)) of the family ({f_k(s)}) has the following properties:

a) (g(s)) is analytic and single-valued in the domain consisting of the half-planes (\sigma>0), (\sigma<0), and the regular points of (f(s)) on the imaginary axis.

b) For (\sigma>0), (g(s)) is represented by a series

[
g(s)=\sum_{n=1}^{\infty} B_n(s)
]

of the same class (B(r,h)) as (f(s)), with bounded coefficients (\beta_{k,\nu}).

c) A convergent expansion holds for (\sigma<0):

[
g(s)=\sum_{n=1}^{\infty} B_{-n}(s),
]

where

[
B_{-k}(s)=\sum_{\nu=1}^{u_{-k}}\beta_{-k,\nu}[e^{xs},\mu_{-k,0},\ldots,\mu_{-k,\nu}],
\quad \text{and}
]

[
|\beta_{-k,\nu}|\leq 1;\quad
\mu_{-k,l+1}\leq \mu_{-k,l};\quad
\mu_{-(k+1),0}-\mu_{-k,u_{-k}}\geq h_1>0;\quad
\mu_{-1,u_{-1}}\geq h_1>0.
]

d) Every isolated singularity of (f(s)) on the imaginary axis is a simple pole of (g(s)) or a regular point of (g(s)).

The proof scheme is the same as for the corresponding theorems in ((^1)), using (9) and (10).

Theorem 3. If, under the conditions of Theorem 2, (g(s)) is a limit function of the family ({f_{k_i}(s)}), where ({k_i}) satisfies (7), then (g(s)) has singular points on the imaginary axis.

Proof scheme. Suppose, on the contrary, that the theorem is false. Then (g(s)) is an entire function. From b), c) of Theorem 2 and (9), (10) it follows that (|g(s)|<A(|s|+1)^r) for (\sigma\geq 1) and (\sigma\leq -1), and (|g(s)|<A_1(|s|+1)^r\times |1-\exp(h_1\sigma)|^{-1}) for (|\sigma|\leq 1). Considering the function (F(s)=g(s){1-\exp[h_1(s-it_0-2\delta)]}{1-\exp[h_1(s-it_0+2\delta)]}), with (\delta<\pi(8h_1)^{-1}), first

in the rectangle ((|\sigma|\leq 1,\ |t-t_0|\leq 2\delta)), and then in the rectangle ((|\sigma|\leq 1,\ |t-t_0|\leq \delta)), it can be shown that (|g(s)|<A_2(|s|+1)^r), and in the strip (|\sigma|\leq 1), and therefore (g(s)) is a polynomial. After this, from b), taking (7) into account, one obtains:
[
g(s)=C_1s^m+\cdots,\qquad C_1\ne0,\qquad m\geq0,
]
and from c), as (\sigma\to-\infty),
[
|g(\sigma)|<C_2|\sigma|^r\exp(h_1\sigma)\to0.
]
The contradiction obtained proves Theorem 3.

Definition. We shall say that (f(s)) has at the point (s_0) a singularity of type (k) ((k\geq0)), if
[
f(s)=a_k(s-s_0)^{-k}+\cdots+a_1(s-s_0)^{-1}+p(s)\log(s-s_0)+f_1(s),
]
where (a_k\ne0), (p(s)) is a polynomial, and (f_1(s)) is regular at the point (s_0).

Theorem 4. If the sum of the series (5) of class (B(r,h)) has on a segment of the imaginary axis of length greater than (2\pi r h^{-1}) only isolated singularities of types 1 and 0, then the coefficients (a_{k,\nu}) are bounded.

The theorem is proved in the same way as the corresponding theorem in (2), using, instead of Pólya’s theorem, Leont’ev’s more general result ((8), pp. 48, 49), and also Theorem 3 of the present paper.

For a series (f(s)) of class (B(r,h)) with bounded coefficients, let us define the family of functions ({f_x(s)}) ((x>0)):
[
f_x(s)=\left[f(s)-\sum_{n=1}^{k(x)} A_n(s)\right]\exp(xs).
]
where (k(x)) is such that
[
\lambda_{m_{k(x)}}<x\leq \lambda_{m_{k(x)}+1}.
]

For the family ({f_x(s)}) and its limit functions (g(s)), Theorem 1, Theorem 2 (with minor changes in c)), and also the following theorem hold.

Theorem 5. If (g(s)=\lim f_{x_j}(s)), then every isolated singularity (i\alpha) of type 1 of (f(s)) on the imaginary axis is a simple pole of (g(s)), and
[
\operatorname{Res}g(i\alpha)=\operatorname{Res}_f(i\alpha)\cdot\lim\exp(i\alpha x_j).
\tag{11}
]

Sketch of the proof. First the theorem is proved for the case of a simple pole of (f(s))—in the same way as in (3). In the general case, consider the function
[
\psi(s)=f(s)-\varphi(s),
]
where the series
[
\varphi(s)=\sum_{n=0}^{\infty} d_n\exp(-ns)
]
is chosen so that (\psi(s)) has at the point (s=i\alpha) a simple pole with the same residue as (f(s)), and (d_n\to0) as (n\to\infty). Then (g(s)) coincides with a certain limit function (h(s)) of the family ({\psi_{x_j}(s)}), which has at the point (i\alpha) a simple pole.

Theorem 6. Let a function (f(s)) of class (B(r,h)), with exponents (\lambda_n), have on a segment of the imaginary axis of length greater than (2\pi(D_\lambda+rh^{-1})) only simple poles at the points (i\alpha_1,i\alpha_2,\ldots,i\alpha_q) ((D_\lambda) denotes the maximal density of ({\lambda_n}), see (4)).

Then:

a) For some (\delta>0), (f(s)) is regular and single-valued in the half-plane (\sigma>-\delta) from which the set (R) of all singular points of (f(s)) on the imaginary axis has been removed.

b) Every isolated point (R) is a simple pole of (f(s)).

c) For every such isolated point (i\alpha), the relation
[
\alpha=m_1\alpha_1+m_2\alpha_2+\cdots+m_q\alpha_q
\tag{12}
]
holds, where (m_1,m_2,\ldots,m_q) are integers.

d) The numbers (m_k) in (12) can be chosen so that

[
m_1+m_2+\cdots+m_q=1.
\tag{13}
]

Proof. a), b), c) are proved in general outline in the same way as in ({}^{3}), using instead of Bernstein’s theorem the result of Leont′ev mentioned above.

Let us prove d). Let (\beta) be a real number which is not a linear combination of (\alpha_1,\ldots,\alpha_q) with rational coefficients. Applying c), already proved, to (f(s-i\beta)), we obtain:

[
\alpha+\beta
=
n_1(\alpha_1+\beta)+n_2(\alpha_2+\beta)+\cdots+n_q(\alpha_q+\beta).
\tag{14}
]

Subtracting (12) from (14), we obtain (n_1+n_2+\cdots+n_q=1) and
[
(n_1-m_1)\alpha_1+\cdots+(n_q-m_q)\alpha_q=0,
]
which together with (12) gives

[
\alpha=n_1\alpha_1+n_2\alpha_2+\cdots+n_q\alpha_q.
]

Theorem 7. Let a function (f(s)) of the class (B(r,h)) have, on the segment (L_1) of length greater than (2\pi r h^{-1}), only isolated singularities of type (\leq k), and on the segment (L) of the imaginary axis of length greater than (4\pi r h^{-1}) only isolated singularities (i\alpha_1,i\alpha_2,\ldots,i\alpha_q).

Then for every isolated singularity (i\alpha) of (f(s)) of type (k) on the imaginary axis, (12) and (13) hold.

Outline of the proof. Integrating (f(s)) (k-1) times, we obtain a series (\Phi(s)) of the class (B(r,h)) which, by Theorem 4, has bounded coefficients. Let (g(s)) be an arbitrary limit function of the family ({\Phi_x(s)}). Every singular point of type (k) of (f(s)) on the imaginary axis is a point of type 1 of (\Phi(s)) and, by Theorem 5, a simple pole of (g(s)). In addition, on (L), (g(s)) has, by virtue of d) of Theorem 2, only simple poles. Application of Theorem 6 to (g(s)) proves Theorem 7.

Remarks. Theorems 6 and 7 (with a certain refinement) make it possible to generalize Theorem 3 from ({}^{7}) and the theorem (4) from ({}^{5}) to broader classes of Dirichlet series. All the theorems from ({}^{6}) can likewise be generalized.

I take this opportunity to express my gratitude to Prof. A. O. Gel′fond and Prof. I. K. Andronov.

Moscow Regional Pedagogical Institute
named after N. K. Krupskaya

Received
19 V 1960

References

({}^{1}) S. Agmon, Ann. École Norm., 66, 263 (1949).
({}^{2}) S. Agmon, Trans. Am. Math. Soc., 74, 444 (1953).
({}^{3}) S. Agmon, Bull. Res. Council of Israel, 8, 4, 385 (1954).
({}^{4}) V. Bernstein, Leçons sur les progrès récents de la théorie des séries de Dirichlet, Paris, 1933.
({}^{5}) B. S. Bronshtein, DAN, 130, No. 4, 719 (1960).
({}^{6}) B. S. Bronshtein, DAN, 131, No. 5, 996 (1960).
({}^{7}) K. Chandrasekharan, S. Mandelbrojt, Ann. of Math., 66, 2 (1957).
({}^{8}) A. F. Leont′ev, Uch. zap. MGU, 146, 3 (1950).