Eine der Riemannschen verwandte Funktionalgleichung
type: Item
instance of: scholarly article
title: Eine der Riemannschen verwandte Funktionalgleichung
zbMATH Open document ID: 0011.06602
DOI: 10.1007/BF01201379
author: Hermann Kober
published in: Mathematische Zeitschrift
publication date: 1935-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/168572
review text: The author discusses the solution of the functional equation \[ F(s)\Gamma^2\left(\frac{s}{2}\right)\left(\frac{\pi}{k}\right)^{-s}=F(1-s)\Gamma^2\left(\frac{1-s}{2}\right)\left(\frac{\pi}{k}\right)^{s-1},\tag{1} \] where \(F(s)\) is representable by a Dirichlet series for sufficiently large values of \(\text{Re}\,(s)\). If \(f(s)\) and \(\Phi(s)\) satisfy \[ f(s)\Gamma\left(\frac{s}{2}\right)\left(\frac{\pi}{k}\right)^{-\frac12 s}=C \Phi(1-s)\Gamma\left(\frac{1-s}{2}\right)\left(\frac{\pi}{k}\right)^{\frac{1-s}{2}},\tag{2} \] where \(C\neq 0\), and \(F(s) = f(s) \Phi(s)\), then \(F(s)\) satisfies (1). The complete solution of (2) has been obtained by Hamburger and Siegel (\textit{H. Hamburger}, Math. Z. 10, 240--254 (1921); 11, 224--245 (1921; JFM 48.1210.03); 13, 283--311 (1922; JFM 48.1213.02)]; \textit{C. L. Siegel} [Math. Ann. 86, 276--279 (1922; JFM 48.1216.01)]), Their result is that \(f(s)\) and \(\Phi(s)\) are linear combinations of Dirichlet's \(L\)-functions \(\pmod k\). The author obtains the particular solution of (1) \[ Z_k(s)=\sum_{n=1}^\infty \mu\{(n,k)\} \varphi\{(n,k)\} d(n)n^{-s}, \] where \(k\) is quadratfrei, \(\mu\) and \(\varphi\) are the functions of Möbius and Euler, \((n, k)\) is the greatest common divisor of \(n\) and \(k\), and \(d(n)\) is the number of divisors of \(n\). For \(k=1\), \(Z_k(s)=\zeta^2 (s)\), and the result is immediate. The proof then goes by induction from \(k\) to \(kp\), where \(p\) is any prime which is prime to \(k\). Vgl. auch das Referat in JFM 61.0324.04.
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 3016377
zbMATH Keywords: Dirichlet series
zbMATH Keywords: related functional equation
zbMATH Keywords: Riemann zeta-function
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3016377