Elementary proof of an estimate on $\zeta$

Question

I saw somewhere that

If $s=\sigma +it$ where $\sigma >0$ and $t\in \mathbb R$,$x\geq |t|/\pi\implies \zeta(s)=\displaystyle \sum_{n\leq x} \frac{1}{n^s}+\frac{x^{1-s}}{s-1}+O(x^{-\sigma})$

• What's the meaning of the asymptotics here ? I don't understand which variables are fixed and which are free.

• Is there a way to prove this with undergraduate means when $t=0$ ?

Answer 1

The proof (for $t=0$ and $s\neq 1$) is given by Euler's summation formula, see Theorem $3.2$ in the book "Introduction to Analytic Number Theory" by Tom Apostol. We just need $x\ge 1$; the asymptotics refers to $x\mapsto \infty$.