MATHEMATICS

G. L. LUNTS

DIRICHLET SERIES WITH A SEQUENCE OF COMPLEX EXPONENTS HAVING ANGULAR DENSITY

(Presented by Academician V. I. Smirnov, 31 I 1963)

For Dirichlet series

\[ f(z)=\sum_{n=1}^{\infty} a_n e^{-\lambda_n z} \tag{1} \]

with positive exponents, theorems are known on the distribution of singularities on the boundary of the half-plane of holomorphy and on the relations between the abscissa of convergence and the abscissa of holomorphy \((^1)\).

A. F. Leont'ev extended these theorems to sequences of Dirichlet polynomials and to Dirichlet series with complex exponents under the assumption that \(\lim_{n\to\infty}\arg\lambda_n=0\) \((^2)\). He also obtained certain theorems for a more general type of sequences \(\{\lambda_n\}\).

In the present article we consider Dirichlet series with complex exponents in the case where the sequence \(\{\lambda_n\}\) has angular density \((^3)\), i.e., where there exists a nondecreasing function \(\sigma(\varphi)\) having the property that for all \(\varphi_1,\varphi_2\) not belonging to a certain exceptional countable set, the density

\[ \lim_{k\to\infty}\frac{k}{|\lambda_{n_k}|} \]

of the sequence \(\{\lambda_{n_k}\}\subset\{\lambda_n\}\), for which \(\varphi_1<\arg\lambda_{n_k}<\varphi_2\), is equal to \(\sigma(\varphi_2)-\sigma(\varphi_1)\). For all \(\varphi\) not belonging to the indicated exceptional set, the function \(\sigma(\varphi)\) is continuous.

\(1^\circ\). The theorems on the distribution of singularities of Dirichlet series with complex exponents are based on a method of defining the domain of convergence of these series \((^4)\) and on the following generalization of the well-known Cramér–Pólya theorem \((^1)\):

If \(|\arg\lambda_n|<\alpha<\pi/2\), the series (1) has a nonempty domain of convergence, the function \(f(z)\) is analytic in the domain \(G\), \(\varphi(z)\) is a function of exponential type in the angle \(-\alpha_1<\arg z<\alpha_2\) \((0<\alpha_1,\alpha_2<\pi/2)\), and \(I\) is a domain inside which lies the conjugate diagram of the function \(\varphi(z)\), then the sum of the series

\[ \sum_{n=1}^{\infty} a_n \varphi(\lambda_n)e^{-\lambda_n z}=F(z) \tag{1} \]

can be analytically continued from the domain of convergence of this series into a domain \(G^*\) such that if \(z_1\in G,\ z_2\in I\), then \(z_1-z_2\in G^*\).

\(2^\circ\). Let the sequence \(\{\lambda_n\}\), having angular density, be such that the set of limit points of the sequence \(\{\arg\lambda_n\}\) is located on the interval \([-\alpha,\alpha]\), where \(\alpha<\pi/2\), and let the condensation index of the sequence \(\{\lambda_n\}\) be

\[ \delta=\lim_{n\to\infty}\frac{1}{|\lambda_n|}\ln\left|\frac{1}{L'(\lambda_n)}\right|, \tag{2} \]

where

\[ L(z)=\prod_{n=1}^{\infty}\left(1-\frac{z^2}{\lambda_n^2}\right), \]

is finite.

Lemma. The function

\[ g(z)=\sum_{n=1}^{\infty}\frac{e^{-\lambda_n z}}{L'(\lambda_n)} \]

is holomorphic outside the angles

\[ \frac{\pi}{2}-\alpha \leq \arg (z-z') \leq \frac{\pi}{2}+\alpha,\qquad -\left(\frac{\pi}{2}+\alpha\right)\leq \arg (z+z') \leq -\left(\frac{\pi}{2}-\alpha\right), \]

where

\[ z'=\pi(S+iC),\qquad S=\int_{\alpha-0}^{\alpha+0}\sin\varphi\,d\sigma(\varphi),\qquad C=\int_{-\alpha-0}^{\alpha+0}\cos\varphi\,d\sigma(\varphi). \]

This lemma, in combination with the above-mentioned theorem of Cramér–Pólya type, makes it possible, by constructing a corresponding function \(\varphi(z)\) taking at the points \(\lambda_n\) the values \(a_n L'(\lambda_n)\), to prove the main theorems.

In the case where the sequence \(\{\lambda_n\}\) satisfies the indicated condition, we shall call an angle of holomorphy of the function \(f(z)\), defined by the series (1), any angle \(|\arg(z-z_0)|<\pi/2-\alpha\) inside which \(f(z)\) is holomorphic, but such that no angle \(|\arg(z-z_1)|<\pi/2-\alpha\), where \(\operatorname{Im} z_1=\operatorname{Im} z_0\), \(\operatorname{Re} z_1<\operatorname{Re} z_0\), has this property.

Theorem 1. If \(z_0\) is the vertex of an angle of holomorphy of the function \(f(z)\), then on the broken line with vertex \(z_0\), consisting of the segment of the ray \(\arg(z-z_0)=\pi/2-\alpha\) of length \(\pi(C/\cos\alpha+S/\sin\alpha)\) and the segment of the ray \(\arg(z-z_0)=-(\pi/2-\alpha)\) of length \(\pi(C/\cos\alpha-S/\sin\alpha)\), there is at least one singular point of the function \(f(z)\).

In the case where the density of the sequence \(\{\lambda_n\}\) is equal to zero (\(S=0\), \(C=0\)), Theorem 1 asserts that the vertex of every angle of holomorphy is a singular point of the function \(f(z)\).

Corollary. If on the “upper” side of the angle of holomorphy of the function \(f(z)\) there is a singular point whose distance from the vertex \(z_0\) of this angle is greater than \(\pi(C/\cos\alpha+S/\sin\alpha)\), then on every segment of length \(\pi(C/\cos\alpha+S/\sin\alpha)\) between this point and the point \(z_0\) there is at least one singular point of the function \(f(z)\). The same assertion is also true for the “lower” side of the angle of holomorphy with respect to segments of length \(\pi(C/\cos\alpha-S/\sin\alpha)\).

Theorem 2. Every ray lying inside an angle of holomorphy \(H\) of the function \(f(z)\), making with the positive direction of the real axis an angle \(\pi/2-\alpha\) and separated from the corresponding side of the angle \(H\) by a distance greater than \(\pi(C\sin\alpha-S\cos\alpha)+\delta\), where \(\delta\) is determined by means of (2), lies entirely, except possibly for a segment of finite length, inside the domain of convergence of the series (1).

An analogous assertion holds for a ray parallel to the other side of the angle \(H\) and separated from this side by a distance greater than \(\pi(C\sin\alpha+S\cos\alpha)+\delta\).

3°. Let \(\Lambda=\{\lambda_n\}\) be any sequence having angular density \((0\leq \arg\lambda_n<2\pi)\); let \(\Lambda_{\varphi_1,\varphi_2}\) be the subsequence of the sequence \(\Lambda\) consisting of those of its terms for which \(\varphi_1\leq \arg\lambda_n<\varphi_2\); let \(R_{\varphi_1,\varphi_2}\) be the series composed of those terms of the series (1) for which \(\lambda_n\in\Lambda_{\varphi_1,\varphi_2}\), and let \(f_{\varphi_1,\varphi_2}(z)\) be the sum of the series \(R_{\varphi_1,\varphi_2}\). That one of the angles of opening \(\pi-2\eta_0\) \((\eta_0>0)\) with bisector directed along the ray \(\arg z=-\varphi\), which belongs to the angles of holomorphy of all the functions \(f_{\varphi,\varphi+\eta}(z)\) for \(0<\eta\leq\eta_0\) and which is boun-

attains the largest domain, denote it by \(H_\varphi^{\eta_0}\). Let \(z=l_\varphi^{(\eta_0)}e^{-i\varphi}\) be its vertex. Analogously, for the functions \(f_{\varphi-\eta,\varphi}(z)\) \((0<\eta\leqslant\eta_0)\) introduce the angle \(H_\varphi^{-\eta_0}\) with vertex at the point \(z=l_\varphi^{(-\eta_0)}e^{-i\varphi}\). Denote
\[ l_\varphi^+=\varlimsup_{\eta_0\to0} l_\varphi^{(\eta_0)},\qquad l_\varphi^-=\varlimsup_{\eta_0\to0} l_\varphi^{(-\eta_0)},\qquad l(\varphi)=\max(l_\varphi^+,l_\varphi^-). \]
Consider the domain \(K\), at all points of which, for every \(\varphi\) \((0\leqslant\varphi<2\pi)\),
\[ x\cos\varphi-y\sin\varphi-l(\varphi)>0. \]
All functions of the form \(f_{\varphi_1,\varphi_2}(z)\), and in particular the function \(f(z)\) itself, are holomorphic in the domain \(K\). Finally, let \(M\) be the closure of the set of singular points of all possible functions of the form \(f_{\varphi_1,\varphi_2}(z)\).

Theorem 3. If the function \(\sigma(\varphi)\) is continuous on the interval \([0,2\pi]\) (in particular, if the density of the sequence \(\{\lambda_n\}\) is equal to zero), then all boundary points of the domain \(K\) belong to the set \(M\).

In the general case: a) all boundary points of the domain \(K\) that do not lie on rectilinear segments contained in this boundary belong to \(M\); b) every rectilinear segment and every polygonal line on the boundary of the domain \(K\) of length \(2\pi D\), where
\[ D=\lim_{n\to\infty}\frac{n}{|\lambda_n|} \]
is the density of the sequence \(\{\lambda_n\}\), contains at least one point of the set \(M\).

Let \(\delta[\varphi_1,\varphi_2)\) be the condensation index of the sequence \(\Lambda_{\varphi_1,\varphi_2}\) (if this sequence is finite, then, by definition, \(\delta[\varphi_1,\varphi_2)=0\)),
\[ \delta_\varphi^+=\varlimsup_{\eta\to0}\delta[\varphi,\varphi+\eta),\qquad \delta_\varphi^-=\varlimsup_{\eta\to0}\delta[\varphi-\eta,\varphi) \]
(it can be shown that always \(\delta_\varphi^+\geqslant0,\ \delta_\varphi^-\geqslant0\)).

Theorem 4. The series (1) converges in the domain whose points, for every \(\varphi\) \((0\leqslant\varphi<2\pi)\), satisfy the condition
\[ x\cos\varphi-y\sin\varphi-m(\varphi)>0, \]
where
\[ m(\varphi)=\max\bigl(l_\varphi^++\delta_\varphi^+,\ l_\varphi^-+\delta_\varphi^-\bigr). \]

Moscow Institute
of Chemical Machine Building

Received
27 I 1963

REFERENCES

\(^{1}\) V. Bernstein, Leçons sur les progrès récents de la théorie des séries de Dirichlet, Paris, 1933.
\(^{2}\) A. F. Leont’ev, Tr. Matemat. inst. im. V. A. Steklova AN SSSR, 39 (1951).
\(^{3}\) B. Ya. Levin, Distribution of Zeros of Entire Functions, Moscow, 1956.
\(^{4}\) G. L. Lunts, Matem. sborn., 10 (52), No. 1–2 (1942).