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Reports of the Academy of Sciences of the USSR
- Volume 161, No. 1
MATHEMATICS
B. S. BRONSTEIN
ON SERIES CLOSE TO DIRICHLET SERIES
(Presented by Academician Yu. V. Linnik on 30 IX 1964)
Let \(\{\lambda_n\}\) be a sequence of positive numbers satisfying the condition:
\[ H > \lambda_{n+1}-\lambda_n > h > 0. \tag{1} \]
Definition 1. A series close to a Dirichlet series, or, briefly, a \(D\)-series, will mean a series
\[ f(z)=\sum_{n=1}^{\infty} a_n\varphi_n(z), \tag{2} \]
if the following conditions are satisfied:
a) The functions \(\varphi_n(z)\) are regular in some bounded domain \(Q\), have no zeros in \(Q\), and satisfy in \(Q\) the relations
\[ \varphi_n(z)=\varphi_0(z)\exp[-\lambda_n z+u(n,z)], \tag{3} \]
\(u(n,z)=o(n)\) uniformly with respect to \(z\);
\[ \varphi_{n+m}(z)=\varphi_n(z)\exp[-(\lambda_{n+m}-\lambda_n)z+v(n,m,z)], \tag{4} \]
\(v(n,m,z)\to 0\) (as \(n\to\infty\) and fixed \(m\)) uniformly with respect to \(z\), and \(v(n,m,z)=o(m)\) uniformly with respect to \(n\) and \(z\).
b) \(Q\) contains some part \(L\) of the boundary of the domain of convergence of series (2).
In view of (1)—(3), \(L\) is situated on a vertical straight line.
Examples of \(D\)-series:
\[ \sum_{n=1}^{\infty} a_n (z-z_0)^{\alpha_n}\exp(-\lambda_n z), \]
where \(\{\lambda_n\}\) satisfies (1), \(\alpha_{n+1}-\alpha_n\to 0\) \((n\to\infty)\),
\[ 0<\overline{\lim}\sqrt[n]{|a_n|}<\infty; \]
a lacunary series in the Chebyshev–Hermite polynomials \(H_{n_k}(iz)\), where
\[ \sqrt{n_{k+1}}-\sqrt{n_k}>h>0; \]
some series in other eigenfunctions of differential equations; the series
\[ \sum_{k=1}^{\infty} a_k z^{\alpha_k}\exp(-\lambda_k z)H_{n_k}(iz) \]
and so on.
Definition 2. A \(D\)-series for which \(L\) lies on the imaginary axis will be called reduced.
A general \(D\)-series is reduced to a reduced one by a linear change of the variable \(z\). From (1)—(3) it follows that for a reduced series
\[ \overline{\lim}\sqrt[n]{|a_n|}=1. \]
In this case (see \((1')\) or \((2')\)) there exists a sequence of positive numbers \(\{q_n\}\) possessing the following properties:
\[ q_{n+1}q_n^{-1}\to 1 \quad (n\to\infty); \tag{5} \]
\[ |a_n|<Cq_n, \tag{6} \]
\[ |a_{n_k}|=q_{n_k} \]
for some infinite sequence of indices \(\{n_k\}\).
\[ \tag{7} \]
Definition 3. Any sequence \(\{n_k\}\) for which there exists \(\{q_n\}\) such that (5), (6), (7) are satisfied will be called a sequence of principal indices of the reduced series.
Some theorems known for Dirichlet series we shall prove here for \(D\)-series.
Theorem 1. Let the sum of the reduced series be regular at the point \(z_0\), \(z_0 \in L\). If \(q_n\varphi_n(z_0)\to 0\) as \(n\to\infty\), then the series converges at the point \(z_0\).
Theorem 2 (on gaps). Let there exist such a sequence of principal indices \(\{n_k\}\) of the reduced series and such a sequence of integers \(\{m_k\}\) \((m_k\to\infty\) as \(k\to\infty)\), that \(a_{n_k+\nu}=0\), \(\nu=1,2,\ldots,m_k\). Then every point of \(L\) is singular for the sum of the series.
The theorems are proved with the aid of the following lemmas.
Lemma 1. Let the sum of the reduced series be regular in some closed domain \(T\), with \(T\subset Q\) and \(T\) containing within itself a segment of the imaginary axis. Then the family of functions \(\{f_n(z)\}\), where
\[ f_n(z)=\left[f(z)-\sum_{j=1}^{n} a_j\varphi_j(z)\right]q_n^{-1}\varphi_n^{-1}(z), \tag{8} \]
is bounded in \(T\).
Proof. Suppose, to the contrary, that the family \(\{f_n(z)\}\) is not bounded in \(T\). Then, if we denote \(p_n=\max_T |f_n(z)|\), there is a sequence of indices \(\{n_l\}\) such that
\[ p_n\to\infty,\quad \text{when } n \text{ runs through } \{n_l\}. \tag{9} \]
Let \(\widetilde f_n(z)=f_n(z)p_n^{-1}\); then \(\max_T |\widetilde f_n(z)|=1\). From \(\{n_l\}\) one can extract a subsequence \(\{n_l'\}\) such that \(\widetilde f_n(z)\to \widetilde g(z)\) uniformly in \(T\), when \(n\) runs through \(\{n_l'\}\); \(\widetilde g(z)\) is regular in \(T\) and
\[ \max_T |g(z)|=1. \tag{10} \]
Let \(T_1=\{T\cap \operatorname{Re} z>\delta\}\), where \(\delta>0\) and is sufficiently small. For \(z\in T_1\) we replace the square bracket in (8) by the remainder of the series (2). Owing to (1), (4), (5), (6), (9), from \(\{n_l\}\) one can, by means of a diagonal process, extract a subsequence \(\{n_l''\}\) along which termwise passage to the limit is possible in the resulting series for \(\widetilde f_n(z)\). A Dirichlet series for \(\widetilde g(z)\) is obtained. But, in view of (9), all its coefficients are equal to zero, and \(\widetilde g(z)\equiv 0\), which contradicts (10). Lemma 1 is proved.
Theorem 1 is a consequence of Lemma 1.
Lemma 2. Under the conditions of Lemma 1, every limit function \(g(z)\) of the family \(\{f_n(z)\}\) is representable in the half-plane \(\operatorname{Re} z>0\) by the Dirichlet series
\[ g(z)=\sum_{m=1}^{\infty} b_m\exp(-\mu_m z), \tag{11} \]
and in the half-plane \(\operatorname{Re} z<0\) by the Dirichlet series
\[ g(z)=b_0+\sum_{m=1}^{\infty} b_{-m}\exp(\mu_{-m}z); \tag{12} \]
\[ H\geq \mu_{\pm(m+1)}-\mu_{\pm m}\geq h>0,\quad \mu_{\pm 1}\geq h>0; \tag{13} \]
\[ |b_{\pm m}|\leq C, \tag{14} \]
where \(H\) and \(h\) are constants from (1), and \(C\) is from (6).
Proof. As in the proof of Lemma 1, one can extract such a subsequence of indices along which termwise passage to the limit in \(T_1\) in (8) is possible, which gives (11), and from it—such a subsequence along which termwise passage to the limit in \(T_2=\{T\cap \operatorname{Re} z<-\delta\}\) is possible, which gives (12).
Lemma 3. If, under the conditions of the preceding lemmas, in (8) \(n\) runs in advance through only some subsequence of principal indices \(\{n_k\}\), then \(g(z)\) has at least one singular point on each segment of the imaginary axis of length greater than \(2\pi h^{-1}\), and in (12) \(|b_0|=1\).
Proof. In (12)
\[ b_0=-\lim (a_n q_n^{-1}) \quad (n\in\{n_k\}); \]
in view of (7), \(|b_0|=1\). If, on the contrary, \(g(z)\) were regular on a segment of the imaginary axis of length greater than \(2\pi h^{-1}\), then, by Pólya’s theorem for Dirichlet series, the boundary of convergence of the series (11) would lie to the left of the imaginary axis. It would follow from (11) and (12) that \(g(z)\) is bounded in the whole plane, i.e. \(g(z)\equiv \mathrm{const}\). But from (11), as \(\operatorname{Re} z\to\infty\), one obtains \(g(z)=0\), while from (12), as \(\operatorname{Re} z\to-\infty\), \(g(z)=b_0\). The contradiction obtained proves Lemma 3.
Proof of Theorem 2. Suppose, to the contrary, that \(f(z)\) can be analytically continued to the left through some segment belonging to \(L\). Then \(f(z)\) is regular in some domain \(T\) of the kind described in Lemma 1. Applying Lemmas 1–3, we obtain in (12) \(|b_0|=1\). But, owing to the omissions in the series (2), all \(b_m\) in (11) vanish, and \(g(z)\equiv0\), which is incompatible with \(|b_0|=1\). The contradiction obtained proves the theorem.
Theorem 3. Every segment belonging to \(L\) and having length greater than \(2\pi h^{-1}\) contains at least one singular point of the sum of the \(D\)-series.
The proof of Theorem 3 is analogous to the preceding one.
Some other theorems on Dirichlet series can also be generalized to \(D\)-series.
Moscow Institute
of Chemical Machine Building
Received
26 IX 1964
REFERENCES
¹ S. Agmon, Ann. Ec. Norm. Sup., 66, 263 (1949).
² B. Ya. Levin, Distribution of Zeros of Entire Functions, 1956.