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 )
.