bounds of Riemann $\zeta(s)$ function on the critical line?

Question

I vaguely remembered that

$$0\leq|\zeta(1/2+i t)|\leq C t^{\epsilon},\qquad t>>1,\epsilon>0$$.

Is this bound correct?

Thanks- mike

Answer 1

In: http://www.math.tifr.res.in/~publ/ln/tifr01.pdf pp.97-99 , it is proven that:

$$\zeta(s) < A(d)t^{1-d}, \text{for } \sigma=\mathrm{Re}(s) \geq d, 0 < d < 1 ; t=\mathrm{Im}(s) \geq 1 .$$

with $A(d) = (1/(1-e) + 1 + 3/e)$.