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 t asymptotics of the Riemann zeta function ζ ( s ) \zeta (s) , s = σ + i t s=\sigma +i t , 0 ≤ σ ≤ 1 0\leq \sigma \leq 1 , t > 0 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 ∑ a b n − s \sum _a^b n^{-s} for certain ranges of a a and b b . In addition, we present precise estimates relating this sum with the sum ∑ c d n s − 1 \sum _c^d n^{s-1} for certain ranges of a , b , c , d a, b, c, d . We also study a two-parameter generalization of the Riemann zeta function which we denote by Φ ( u , v , β ) \Phi (u,v,\beta ) , u ∈ C u\in \mathbb {C} , v ∈ C v\in \mathbb {C} , β ∈ R \beta \in \mathbb {R} . Generalizing the methodology used in the study of ζ ( s ) \zeta (s) , we derive asymptotic formulae for Φ ( u , v , β ) \Phi (u,v, \beta ) .