Let for Re(s) >0 and $x\geq 1$,
$\zeta(s)= \sum_{n\leq x} \frac{1} {n^s} +\frac{x^{1-s}} {1-s}$+{x}$/x^s$-s $\int_{x}^{\infty}${u}$/u^{s+1} du$, where {x} = x-[x] , where [x] is largest integer less than or equal to x. be a known equation. ---(1)
Use this equation prove that for $t\geq 2$, $|\zeta (\sigma +i t) | \ll$ log t if $\sigma $=1.
I put $s = 1+it $ in the equation 1 to get RHS is less than or equal to $\sum_{n\leq x} 1/n $ + 1/t+ {x} /x -(1+it) $\int_{x}^{\infty} ${u} / $u^{2+it} du$ . but I am unable to get the result which is required.
Can you please help.