On solutions to Riemann's functional equation
The authors prove the following: Let \(G(s)\) be an entire function of finite order, \(P(s)\) a polynomial, \(f(s) = G(s)/P(s)\), with \[ f(s) = \sum_{n=1}^\infty a_n n^{-s}\qquad(\text{Re}\, s = \sigma > 1) \] converging absolutely. Let \(f(s)\) and \(g(s)\) satisfy the functional equation \[ f(s)\Gamma({\textstyle{1\over 2}}s)\pi^{-s/2} = g(2k-s)\Gamma(k - {\textstyle{1\over 2}}s)\pi^{-(k-s/2)}, \] where \[ g(2k-s) = \sum_{n=1}^\infty b_n n^{s-2k} \] is absolutely convergent for \(\sigma <\alpha < 2k-1\). Then for \(k>0\), \(f(s) = a_1\zeta(s)\) for \(k = {1\over 2}\) (\(\zeta(s)\) is the Riemann zeta-function), but \(f(s) \equiv 0\) for \(k\not={1\over 2}\). This result generalizes the well-known case \(k = {1\over 2}\), which is a theorem of \textit{H. Hamburger} [Math. Z. 10, 240--254 (1921; JFM 48.1210.03)]. The authors' proof is modelled after the classical proof of \textit{C. L. Siegel} of Hamburger's theorem [Math. Ann. 86, 276--279 (1922; JFM 48.1216.01)].
scientific article; zbMATH DE number 1455272