Is there a zeta function(with a Dirichlet series) having known roots off the critical line?

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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.




roots riemann-zeta dirichlet-series




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    up vote
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    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.




    roots riemann-zeta dirichlet-series




    share|cite|improve this question





















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      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.




      roots riemann-zeta dirichlet-series




      share|cite|improve this question











      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.





      roots riemann-zeta dirichlet-series





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      asked Aug 6 at 23:09









      crow

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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.






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          • It isnt actually a dirichlet series. math.stackexchange.com/questions/2170370/…
            – crow
            Aug 7 at 15:42










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          1 Answer
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          1 Answer
          1






          active

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          active

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          up vote
          1
          down vote



          accepted










          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.






          share|cite|improve this answer





















          • It isnt actually a dirichlet series. math.stackexchange.com/questions/2170370/…
            – crow
            Aug 7 at 15:42














          up vote
          1
          down vote



          accepted










          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.






          share|cite|improve this answer





















          • It isnt actually a dirichlet series. math.stackexchange.com/questions/2170370/…
            – crow
            Aug 7 at 15:42












          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          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.






          share|cite|improve this answer













          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.







          share|cite|improve this answer













          share|cite|improve this answer



          share|cite|improve this answer











          answered Aug 7 at 4:23









          Alex R.

          23.7k12352




          23.7k12352











          • It isnt actually a dirichlet series. math.stackexchange.com/questions/2170370/…
            – crow
            Aug 7 at 15:42
















          • It isnt actually a dirichlet series. math.stackexchange.com/questions/2170370/…
            – crow
            Aug 7 at 15:42















          It isnt actually a dirichlet series. math.stackexchange.com/questions/2170370/…
          – crow
          Aug 7 at 15:42




          It isnt actually a dirichlet series. math.stackexchange.com/questions/2170370/…
          – crow
          Aug 7 at 15:42












           

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