About the zeros of $f_n(z)=\sum_{k=1}^n k^{-z}$.

Question

Let $z$ be a complex number. Consider $f_n(z)=\sum_{k=1}^n k^{-z}$.

Now I wonder : Are there infinitely many positive integer $n$ such that there exists a $z$ with $f_n(z)=0$ and $Re(z)>1$ ?

I know that such $z$ exists for $n=13$.

I don't understand what the "I know it is true for $n=13$" means in relation with your question above. Should it be a "I know that such a $z$ exists for $n=13$" or what? — Daniel Robert-Nicoud, Dec 19 '13 at 23:54

score 4 · Accepted Answer · edited Feb 09 '23 at 17:01

4

I strongly suggest you read this article by P. Borwein et al. (Experimental math, 2007), as well as this followup by Gonek and Ledoan.

edited Feb 09 '23 at 17:01

Glorfindel

answered Dec 19 '13 at 23:56

Igor Rivin

1 Answers1