A zero-free region for the fractional derivatives of the Riemann zeta function
Keywords:
Riemann zeta function, fractional derivatives, zero-free regionsAbstract
For any 
, we denote by ![D_s^{\alpha}[\zeta(s)]](http://nzjm.math.auckland.ac.nz/images/math/c/c/2/cc2f21d856c7ef15e90cad77ce677333.png)
the α-th Grunwald-Letnikov fractional derivative of the Riemann zeta function ζ(s). For these derivatives we show:
![D_s^\alpha[\zeta(s)]\ne 0](http://nzjm.math.auckland.ac.nz/images/math/d/2/3/d23b76c254f1e58dd5d332e217afa86c.png)
inside the region | s − 1 | < 1. This result, the first of its kind, is proved by a careful analysis of integrals involving Bernoulli polynomials and bounds for fractional Stieltjes constants.
