Eine der Riemannschen verwandte Funktionalgleichung
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.
scientific article; zbMATH DE number 3016377