# Generalizations of Hamburger's Theorem

(Despite the name, the theorem in question is not a joke nor is it a statement about a delicious food).

An old theorem of Hans Hamburger from 1921, as stated in Marvin Knopp's paper "On Dirichlet series satisfying Riemann's functional equation" (Knopp took the statement from a work of Hecke), asserts that the Riemann zeta function $$\zeta$$ is determined by its functional equation, in the following sense: suppose $$R(s) = \pi^{-s} \Gamma(s) \varphi(s)$$, where $$\varphi(s)$$ is a meromorphic function. If $$\varphi(s)$$ satisfies:

1. $$P(s) \varphi(s)$$ is entire for some polynomial $$P$$;
2. $$\varphi(s)$$ satisfies the functional equation $$R(s) = R(1/2 - s)$$; and
3. a) Both $$\varphi(s)$$ and $$\varphi(s/2)$$ can be expressed as a Dirichlet series convergent somewhere, so that $$\varphi(s) = \sum_{n \geq 1} b(n) n^{-2s}$$ b) $$\varphi(s)$$ can be expressed as a Dirichlet series, and the only pole of $$\varphi$$ is at $$s = 1/2$$.

Then both 1, 2, 3a) and 1,2, 3b imply that $$\varphi$$ is a scalar multiple of $$\zeta(2s)$$.

Has Hamburger's theorem been generalized to other functional equations appearing for well-known $$L$$-functions, such as Dirichlet $$L$$-functions?

• Hamburger is an adjective in German. It means "from [the city of] Hamburg". It is a common word and a common name in German. So I recommend deleting the first sentence as irrelevant. Commented Jul 21 at 6:36
• @GHfromMO, my first instinct was to wonder whether it was a misnomer for the ham sandwich theorem. Commented Jul 21 at 7:42
• @GHfromMO: In English, the food meaning is far more common and more salient, even for those of us well aware that it’s also a name and adjective in German; I had exactly the same initial reaction as Peter Taylor. Commented Jul 21 at 12:21
• @PeterLeFanuLumsdaine In my opinion, the post would be better without the first sentence, especially that the second sentence clears all doubt by mentioning the full name "Hans Hamburger". Commented Jul 21 at 12:35

Hamburger's theorem has been generalized in various ways to automorphic $$L$$-functions (of arbitrary degree). Such generalizations are called "converse theorems", and they play a central role in the Langlands program. See for example this paper by Cogdell and Piatetski-Shapiro, especially the historical notes on the first two pages. See also these notes by Cogdell.