On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function

We present several formulae for the large $t$ asymptotics of the Riemann zeta function $\zeta (s)$, $s=\sigma +it$, $0\le \sigma \le 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations for the sum ${\sum }_a^b{n}^{-s}$ for certain ranges of $a$ and $b$. In addition, we present precise estimates relating this sum with the sum ${\sum }_c^d{n}^{s-1}$ for certain ranges of $a,b,c,d$. We also study a two-parameter generalization of the Riemann zeta function which we denote by $\mathrm{\Phi }(u,v,\beta )$, $u\in \mathbb{C}$, $v\in \mathbb{C}$, $\beta \in $\mathbb{R}$$\mathrm{.}Generalizingthemethodologyusedinthestudyof$\zeta(s)$, we derive asymptotic formulae for $\mathrm{\Phi }(u,v,\beta )$.