# §25.9 Asymptotic Approximations

If $x\geq 1$, $y\geq 1$, $2\pi xy=t$, and $0\leq\sigma\leq 1$, then as $t\to\infty$ with $\sigma$ fixed,

 25.9.1 $\mathop{\zeta\/}\nolimits\!\left(\sigma+it\right)=\sum_{1\leq n\leq x}\frac{1}% {n^{s}}+\chi(s)\sum_{1\leq n\leq y}\frac{1}{n^{1-s}}+\mathop{O\/}\nolimits\!% \left(x^{-\sigma}\right)+\mathop{O\/}\nolimits\!\left(y^{\sigma-1}t^{\frac{1}{% 2}-\sigma}\right),$

where $s=\sigma+it$ and

 25.9.2 $\chi(s)=\pi^{s-\frac{1}{2}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}-% \tfrac{1}{2}s\right)/\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}s\right).$

If $\sigma=\frac{1}{2}$, $x=y=\sqrt{t/(2\pi)}$, and $m=\left\lfloor x\right\rfloor$, then (25.9.1) becomes

 25.9.3 $\mathop{\zeta\/}\nolimits\!\left(\tfrac{1}{2}+it\right)=\sum_{n=1}^{m}\frac{1}% {n^{\frac{1}{2}+it}}+\chi\left(\tfrac{1}{2}+it\right)\sum_{n=1}^{m}\frac{1}{n^% {\frac{1}{2}-it}}+\mathop{O\/}\nolimits\!\left(t^{-1/4}\right).$

For other asymptotic approximations see Berry and Keating (1992), Paris and Cang (1997); see also Paris and Kaminski (2001, pp. 380–389).