Is there a zeta function(with a Dirichlet series) having known roots off the critical line?
I thought there was something like the Hilldebrand-Davis zeta function or something like that, but I can't remember exactly what it was.
2026-03-25 03:07:30.1774408050
Is there a zeta function(with a Dirichlet series) having known roots off the critical line?
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You're probably thinking of the Davenport-Heilbronn zeta function. The original paper is:
On the Zeros of Certain Dirichlet Series - Davenport, Heilbronn.
You'll likely find it useful to read section 10.25 (page 282) of Titmarsh's book for a slightly easier example of proving that a certain zeta function has a zero to the right of the critical strip. A more involved proof (see first link) can show that the Davenport Heilbronn zeta function has a root inside the critical strip, but off the critical strip. There's also nice numerical computations here:
Zeros of the Davenport-Heilbronn Counterexample - Balanzario, et. al.