Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1020.11052
Culp-Ressler, Wendell; Flood, Kevin; Heath, Ann; de Azevedo Pribitkin, Wladimir
On solutions to Riemann's functional equation. (English)
[J] Ramanujan J. 4, No.1, 5-9 (2000). ISSN 1382-4090; ISSN 1572-9303

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 $$ f(s) = \sum_{n=1}^\infty a_n n^{-s}\qquad(\Re s = \sigma > 1) $$ 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 {\it H. Hamburger} [Math. Z. 10, 240-254 (1921; JFM 48.1210.03)]. The authors' proof is modelled after the classical proof of {\it C. L. Siegel} of Hamburger's theorem [Math. Ann. 86, 276-279 (1922; JFM 48.1216.01)].
[Aleksandar Ivić (Beograd)]

MSC 2000:: *11M06 Riemannian zeta-function and Dirichlet L-function

Keywords: Riemann's functional equation; zeta-function; Hamburger's theorem

Citations: JFM 48.1210.03; JFM 48.1216.01

Cited in: Zbl 0958.11060

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.
	Elementary number theory. Primes, congruences, and secrets.

Advanced Search

Zentralblatt MATH Berlin [Germany]