Application of Continued Fractions to the Computation of the Riemann Zeta Function
Abstract Generated abstract
The paper examines the use of continued fractions for computing the Riemann zeta function, the generalized zeta function, and related series involving zeta values. Using positivity properties of certain Hankel determinants formed from the terms of these series, it derives continued fraction representations with positive partial numerators for the Lerch type function and specializes them to the Riemann zeta function. The article also applies the same approach to series expansions for the psi function and the logarithm of the gamma function, giving corresponding convergents and remainder estimates.
Full Text
APPLICATION OF CONTINUED FRACTIONS TO THE COMPUTATION OF THE RIEMANN ZETA FUNCTION
V. E. KORNILOV
(Presented by the Department of Higher Mathematics)
The article studies the question of applying continued fractions to the computation of the Riemann zeta function (7), its generalizations, and the computation of functions represented by series whose terms are the corresponding zeta functions. For the latter series, such generalizations are obtained that can also be computed by means of continued fractions.
1. The generalized zeta function is defined by the following equalities ([1], pp. 40–43; [2], pp. 164, 9):
\[ \zeta(s,v)=\sum_{n=0}^{\infty}(v+n)^{-s},\quad v\ne 0,-1,-2,\ldots,\quad s>1; \tag{1} \]
\[ \zeta(s,v)=\frac{1}{\Gamma(s)}\sum_{n=0}^{\infty}\left[\int_{0}^{\infty} e^{-nt}e^{-vt}t^{s-1}\,dt\right],\quad v>0,\quad s>0; \tag{2} \]
\[ \left\{ \begin{aligned} \zeta(k,v)&=\frac{1}{2v^k}+\int_{-\pi/2}^{\pi/2} \frac{\cos^{k-2}t\,\sin(kt)\,dt}{v^{k-1}\operatorname{th}(\pi v \operatorname{tg} t)},\quad v>0,\\ k&=2,3,\ldots;\quad \varphi_m=\cos^m t(\sin mt+\cos mt) \end{aligned} \right. \tag{3} \]
([3], p. 211). Let
\[ (v+n)^{-s}=c_n,\quad \zeta(n,v)=d_n, \tag{4} \]
then, by virtue of equalities (1)–(4), the theorem ([3], p. 211), and also ([4], p. 226; [5], p. 153; [6], problem 943), the determinants
\[ \left| \begin{array}{cccc} c_{p-1} & \cdot & \cdot & c_{p+m-2}\\ \cdot & \cdot & \cdot & \cdot\\ c_{p+m-2} & \cdot & \cdot & c_{p+2m-3} \end{array} \right|, \quad \left| \begin{array}{cccc} d_{p+1} & \cdot & \cdot & d_{p+m}\\ \cdot & \cdot & \cdot & \cdot\\ d_{p+m} & \cdot & \cdot & d_{p+2m-1} \end{array} \right|; \quad p,m=1,2,\ldots \tag{5} \]
are positive.
2. The functions ([1], pp. 42, 47)
\[ \Phi(z,s,v)=\sum_{n=0}^{\infty}(v+n)^{-s}z^n,\quad |z|<1,\quad v\ne 0,-1,\ldots; \tag{6} \]
\[ \zeta(s)=\zeta(s,1)=\Phi(1,s,1)=\frac{1}{1-2^{1-s}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^s},\quad s>0. \tag{7} \]
on the basis of the positive values of the first determinants (5), are represented by continued fractions with positive partial numerators. Thus we obtain the following convergents for the functions (6) and (7), and also have the corresponding estimate of the remainder term in modulus (see [7]; [8], pp. 27–30).
\[ \left\{ \begin{aligned} \Phi(z,s,v)&=P_{2n}(z):Q_{2n}(z)+R_{2n}(z)=\\ &= \left| \begin{array}{cccccc} 0&S_0z^{\,n-1}&\cdots&S_{n-1}\\ c_0&c_1&&c_n\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ c_{n-1}&c_n&\cdots&c_{2n-1} \end{array} \right| : \left| \begin{array}{ccccc} z^n&\cdots&1\\ c_0&\cdots&c_n\\ \cdot&\cdot&\cdot&\cdot\\ c_{n-1}&\cdots&c_{2n-1} \end{array} \right| +\\ &\quad +R_{2n}(z);\quad n=1,2,\ldots;\quad S_n=\sum_{k=0}^{n}(v+k)^{-s}z^k; \end{aligned} \right. \tag{8} \]
\[ \zeta(s)=\frac{1}{1-2^{1-s}} \left[ \frac{P_{2n}(-1)}{Q_{2n}(-1)}+R_{2n}(-1) \right], \quad v=1,\quad n=1,2,\ldots; \tag{9} \]
\[ |R_{2n}(z)|< \left| \frac{P_{2n+1}(z)}{Q_{2n+1}(z)} - \frac{P_{2n}(z)}{Q_{2n}(z)} \right|, \quad \operatorname{Re}z\leqslant 0. \tag{10} \]
- If in the equality ([1], p. 60)
\[ \psi(1+z)=-\gamma+\sum_{n=1}^{\infty} \left(\frac1n-\frac{1}{z+n}\right), \quad \gamma=0.577\ldots \tag{11} \]
we use the expansion
\[ \frac1\kappa-\frac{1}{z+\kappa} = \frac{z}{\kappa^2} - \frac{z^2}{\kappa^3} + \frac{z^3}{\kappa^4} -\ldots,\quad |z|<\kappa, \]
then we obtain
\[ \left\{ \begin{aligned} \psi(1+z)&=-\gamma+ \sum_{n=1}^{\kappa-1} \left(\frac1n-\frac{1}{z+n}\right)+\\ &\quad+\sum_{n=2}^{\infty}(-1)^n\zeta(n,\kappa)z^{n-1}, \quad |z|<\kappa. \end{aligned} \right. \tag{12} \]
Integrating equality (12), we find ([1], p. 30, (1))
\[ \ln\Gamma(1+z) = -\gamma z+ \sum_{n=1}^{\kappa-1} \left[ \frac zn-\ln\left(1+\frac zn\right) \right]+ \tag{13} \]
\[ +\sum_{n=2}^{\infty}(-1)^n\zeta(n,\kappa)\frac{z^n}{n}, \quad |z|<\kappa. \]
The power series (12) and (13), by virtue of the positive values of the second determinants (5), are represented by continued fractions with positive partial numerators. In this way, according to equality (8), we obtain analogous convergents for the series (12) and (13) and, according to inequality (10), an analogous estimate of the remainder term ([9], problem 1220).
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