# Non-holomorphic L-functions

In some old notes, I found the following conjecture:

Let $\mathbb T$ denote the unit circle and let $\chi:{\mathbb N} \to {\mathbb T}$ be fully multiplicative. Then the L-series $$L(\chi,s)=\sum_{n=1}^\infty\frac{\chi(n)}{n^s},$$ defined for ${\rm Re}(s)>1$, cannot be extended to a holomorphic function on $\{ {\rm Re}(s)\ge 0\}$.

Who formulated this conjecture? What related results are known?

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What does "fully multiplicative" mean? completely multiplicative? –  Kevin Smith Mar 23 '12 at 9:56
The second question is easy: it is false. See mathoverflow.net/questions/15176/…. The answer to the first question is "We'll never know". Indeed, how are you going to prove that some lost manuscript of Archimedes (or some private letter of Euler, or...) did not contain the full discussion already? –  fedja Mar 23 '12 at 10:08
@fedja: Thanks a lot! –  doug Mar 23 '12 at 11:06