B. S. BRONSTEIN

ON THE SOLUTION OF EQUATIONS OF RIEMANN TYPE IN THE CLASS OF DIRICHLET SERIES

(Presented by Academician P. S. Aleksandrov, 11 IX 1959)

§ 1. Introduction.

Hamburger proved that every ordinary Dirichlet series

\[ f(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}, \]

satisfying the Riemann functional equation

\[ f(s)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right) = f(1-s)\pi^{-(1-s)/2}\Gamma\left(\frac{1-s}{2}\right) \tag{1} \]

and certain additional assumptions concerning the regularity of \(f(s)\), has the form \(f(s)=c\zeta(s)\), where \(\zeta(s)\) is the Riemann function and \(c\) is a constant. Hamburger also gave some generalizations of his theorem to the case of the functional equation for \(L\)-series. Bochner, Chandrasekhar, and Mandelbrojt \((^{1,2})\) considered the equation of a somewhat more general form

\[ f(s)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right) = g(\delta-s)\pi^{-(\delta-s)/2}\Gamma\left(\frac{\delta-s}{2}\right) \tag{2} \]

and showed that, for two given sequences \(\{\lambda_n\}\) and \(\{\mu_n\}\),

\[ 0<\lambda_1<\lambda_2<\cdots,\qquad \lambda_n\to\infty; \]

\[ 0<\mu_1<\mu_2<\cdots,\qquad \mu_n\to\infty, \]

one of which has finite upper density, and for fixed \(\delta>0\), equation (2) has a finite number of linearly independent solutions of the form

\[ \left\{ f(s)=\sum_{n=1}^{\infty}\frac{a_n}{\lambda_n^s},\quad g(s)=\sum_{n=1}^{\infty}\frac{b_n}{\mu_n^s} \right\}. \]

Moreover, under additional assumptions concerning \(\{\lambda_n\}\) or \(\{\mu_n\}\), these authors proved several theorems on the form of the solutions.

In the present paper we consider the class of equations of the form

\[ f(s)P(s)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right) = g(1-s)Q(s)\pi^{-(1-s)/2}\Gamma\left(\frac{1-s}{2}\right), \tag{3} \]

where \(\Gamma(s)\) is Euler’s function; \(P(s)\) and \(Q(s)\) are arbitrary fixed polynomials without common divisors, \(P(s)\) is of degree \(m\), and \(Q(s)\) is of degree \(l\).

Definition. Let two sequences \(\{\lambda_n\}\) and \(\{\mu_n\}\) be given,

\[ 0<\lambda_1<\lambda_2<\cdots,\quad \lambda_n\to\infty;\qquad 0<\mu_1<\mu_2<\cdots,\quad \mu_n\to\infty \quad \text{as } n\to\infty. \]

We shall call a pair of functions \(\{f(s),g(s)\}\) a solution of equation (3), belonging to the label \(\{\lambda_n,\mu_n\}\), if:

1) There exists a sequence of complex numbers \(\{a_n\}\), among which at least one \(a_n\ne0\), and a sequence \(\{b_n\}\), for which

at least one \(b_n\ne 0\), such that the functions \(f(s)\) and \(g(s)\) are representable in the form of general Dirichlet series:

\[ f(s)=\sum_{n=1}^{\infty}\frac{a_n}{\lambda_n^s},\qquad g(s)=\sum_{n=1}^{\infty}\frac{b_n}{\mu_n^s}, \]

where both series have abscissas of convergence \(\sigma_f\) and \(\sigma_g\):

\[ -\infty \leq \sigma_f<\infty,\qquad -\infty \leq \sigma_g<\infty \qquad (s=\sigma+i\tau). \]

2) There exists a function \(\chi(s)\), analytic and single-valued in the domain \(|s|>R\) (where \(R\) is some constant), tending to zero as \(\tau\to\infty\) uniformly in any bounded strip \(\sigma_1\leq \sigma\leq \sigma_2\), and, for sufficiently large \(\sigma\), the function \(\chi(s)\) coincides with the right-hand side of equation (3), while for sufficiently small \(\sigma\) it coincides with the left-hand side of equation (3), i.e.

\[ \chi(s)=f(s)P(s)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right) \qquad \text{for } \sigma>\alpha, \]

\[ \chi(s)=g(1-s)Q(s)\pi^{-(1-s)/2}\Gamma\left(\frac{1-s}{2}\right) \qquad \text{for } \sigma<\beta. \]

§ 2. Main theorems on solutions of equation (3).

Theorem 1. Equation (3) has a solution in the class of ordinary Dirichlet series

\[ \left\{f(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s},\quad g(s)=\sum_{n=1}^{\infty}\frac{b_n}{n^s}\right\} \]

in the following and only in the following cases:

1) \(P(s)=s(s+1)(s+2)\ldots(s+2k-1);\ Q(s)=\operatorname{const}.\)

2) \(P(s)=\operatorname{const};\ Q(s)=(s-1)(s-2)\ldots(s-2k)\) \((k\) is a positive integer\()\).

Moreover, in case 1) all solutions of equation (3) have the form \(f(s)=c\,\zeta(s+2k)\), while in case 2) all solutions of (3) have the form \(f(s)=c\,\zeta(s-2k)\).

Theorem 2. If a solution

\[ \left\{f(s)=\sum_{n=1}^{\infty}\frac{a_n}{\lambda_n^s},\quad g(s)=\sum_{n=1}^{\infty}\frac{b_n}{\mu_n^s}\right\} \]

of equation (3) has the property that \(\mu_n=m^{(n)}\mu\), where \(m^{(n)}\) are integers, then this solution has the form

\[ f(s)=\alpha^s\sum_{\lambda<1}a_\lambda\left[\zeta(s+k,\lambda)+(-1)^k\zeta(s+k,1-\lambda)\right]\qquad(\alpha>0), \]

where \(k\) is an integer, and \(\lambda\) runs through a finite set of values. \(\zeta(s,0)\) is, by definition, considered equal to \(\zeta(s)\). The polynomials \(P(s),Q(s)\) must then have a special form (determined by the number \(k\)).

Theorem 3. Let two strictly increasing sequences of positive numbers \(\{\lambda_n\}\), \(\{\mu_n\}\), tending to infinity, be given, and suppose that

\[ \underline{\lim}\,(\mu_{n+1}-\mu_n)\geq h>0. \]

Denote

\[ D^+=\overline{\lim}\,\frac{n}{\lambda_n}\qquad(D^+<\infty). \]

Let \(\varepsilon\) be any positive number. Then the number of linearly independent solutions of equation (3) belonging to the label \(\{\lambda_n,\mu_n\}\) does not exceed the number of \(\lambda_n\)’s lying in any interval \((a,b)\) of the positive axis of length \(b-a=D^++\varepsilon\).

Theorem 4. Let

\[ \left\{f(s)=\sum_{n=1}^{\infty}\frac{a_n}{\lambda_n^s},\quad g(s)=\sum_{n=1}^{\infty}\frac{b_n}{\mu_n^s}\right\} \]

be a solution of equation (3). Suppose

\[ h_\mu=\underline{\lim}\,(\mu_{n+1}-\mu_n)>0 \]

and let \(r,k\) be positive integers such that

\[ \lambda_{r+k+1}-\lambda_r>2h_\mu^{-1}. \]

Then \(k \geq 2\), and each \(\lambda_n\) has the form

\[ \lambda_n=m_1^{(n)}\lambda_{r+1}+m_2^{(n)}\lambda_{r+2}+\cdots+m_k^{(n)}\lambda_{r+k}, \]

where \(m_i^{(n)}\) are integers.

Further, if \(q\) is the least integer such that \(\lambda_{q+1}>h_\mu^{-1}\), then each \(\lambda_n\) has the form
\(\lambda_n=p_1^{(n)}\lambda_1+p_2^{(n)}\lambda_2+\cdots+p_q^{(n)}\lambda_q\), where \(p_i^{(n)}\) are integers.

From Theorem 4, with the aid of Theorems 1 and 2, one can obtain some corollaries. We give one of them.

Corollary. If the solution \(\{f(s),g(s)\}\) of equation (3) is such that
\(\lim(\mu_{n+1}-\mu_n)=h_\mu>0\) and \(\lambda_1<h_\mu^{-1}<\lambda_2\), then
\(f(s)=\alpha^s\zeta(s+2k)[2^{1-(s+2k)}-1]\), where \(k\) is an integer and \(\alpha\) is a positive number.

§ 3. Theorems 1, 2, 3, 4 are derived from the following theorem:

Theorem A. If

\[ \left\{ f(s)=\sum_{n=1}^{\infty}\frac{a_n}{\lambda_n^s},\quad g(s)=\sum_{n=1}^{\infty}\frac{b_n}{\mu_n^s}\right\} \]

form a solution of equation (3), then the equality

\[ \sum_{n=1}^{\infty}\sum_{k=0}^{m} \frac{\alpha_k a_n \lambda_n^{2k} t^l}{(t^2+\lambda_n^2)^{k+1}} +K(t) = \sum_{j=0}^{l}\beta_j t^j \frac{d^j}{dt^j}F(t). \tag{A} \]

holds.

Here \(K(t)\) is a function defined and regular for all \(t\) for which \(\log t\) is defined, and moreover \(|K(t)|=O(|t|^{1-\eta})\) uniformly in the angle \(|\arg t|\leq\theta_0\) for any \(\theta_0\) and some \(\eta>0\);

\[ F(t)=\sum_{n=1}^{\infty}b_n e^{-2\pi\mu_n t}\quad (\text{for } \operatorname{Re} t>0); \]

\(\alpha_k\) are coefficients depending on \(P(s)\); \(\beta_j\) are coefficients depending on \(Q(s)\).

Remark. If the series in the left-hand side of (A) does not converge absolutely, then one must first make in (3) the substitution \(s=s'+2p\), where \(p\) is a sufficiently large integer.

For the proof of Theorems 1 and 2 it is necessary to take from both sides of (A) the finite difference of order \(l\) and then use the periodicity of the right-hand side of the equality obtained.

For the proof of Theorem 3 it is necessary to regard (A) as a differential equation with respect to \(F(t)\) and to apply Mandelbrojt’s theorem \((^3)\) on the singularities of the series \(F(t)\) on the segment of the axis of convergence. A stronger form of the theorem is obtained if one applies another theorem of Mandelbrojt \((^4)\). Finally, for the proof of Theorem 4 a theorem of the type of Agmon’s theorem \((^5)\), but somewhat more general, is needed.

Definition. We shall say that the function \(F(s)\) has at the point \(s_0\) a singularity of type \(k\) if the equality

\[ F(s)=\frac{r_k}{(s-s_0)^k}+\frac{r_{k-1}}{(s-s_0)^{k-1}}+\cdots+\frac{r_1}{s-s_0} +q(s)\ln(s-s_0)+r_0+F_1(s), \]

holds, where \(r_k\ne0\) \((k\geq0)\); \(q(s)\) is a polynomial; \(F_1(s)\) is regular and vanishes at the point \(s=s_0\).

Theorem B. Let \(F(s)\) be a function representable for \(\sigma>\sigma_c\) by the series

\[ F(s)=\sum_{n=1}^{\infty}p_n(s)e^{-\mu_n s}, \]

where \(\lim(\mu_{n+1}-\mu_n)\geq h>0\);
\(p_n(s)=a_n^{(l)}s^l+a_n^{(l-1)}s^{l-1}+\cdots+a_n^0\) is a polynomial of degree not exceeding \(l\);
\(a_n^{(j)}=o(a_n^{(m)})\) for \(j\ne m\), \(n\to\infty\). Suppose that on some segment \(L\) of the axis of convergence of length greater than \(4\pi h^{-1}\)

there are only isolated singularities of \(F(s)\)—the points \(\sigma_c+\alpha_1 i\), \(\sigma_c+\alpha_2 i,\ldots,\sigma_c+\alpha_q i\), and some segment \(L_1\) of the axis of convergence of length greater than \(2\pi h^{-1}\), not containing the point \(s=\sigma_c\), contains only isolated singularities of \(F(s)\) of type \(k\) or of lower type.

Then, for any singular point of type \(k\) on the axis of convergence \(s=\sigma_c+\alpha i\), the equality
\[ \alpha=m_1^{(\alpha)}\alpha_1+m_2^{(\alpha)}\alpha_2+\cdots+m_q^{(\alpha)}\alpha_q \]
holds, where \(m_j^{(\alpha)}\) are integers. Moreover, if \(\sigma_c+\alpha_n i\) is a singularity on \(L\) of type lower than \(k\), then one may assume that \(m_n^{(\alpha)}=0\).

I take this opportunity to express my deep gratitude to Prof. A. O. Gelfond for a number of valuable pieces of advice and to Prof. I. K. Andronov for his attention to my work.

Moscow Regional Pedagogical Institute
named after N. K. Krupskaya

Received
11 IX 1959

REFERENCES

¹ S. Bochner, K. Chandrasekharan, Ann. of Math., 63, 336 (1956).
² K. Chandrasekharan, S. Mandelbrojt, Ann. of Math., 66, No. 2 (1957).
³ S. Mandelbrojt, The Rice Inst. Pamphlet, 31, 159 (1944).
⁴ S. Mandelbrojt, Adherent Series, Regularization of Sequences, Applications, 1955.
⁵ S. Agmon, Bull. Res. Council Israel, 3, No. 4 (1954).