In the present paper, we give bounds for the Tornheim double zeta function
T
(
s
1
,
s
2
,
s
3
)
T(s_1,s_2,s_3)
when
|
t
1
|
,
|
t
2
|
,
|
t
3
|
≥
1
\lvert t_1 \rvert , \lvert t_2 \rvert , \lvert t_3 \rvert \ge 1
,
|
t
1
+
t
2
|
,
|
t
2
+
t
3
|
,
|
t
3
+
t
1
|
≥
1
\lvert t_1 + t_2 \rvert , \lvert t_2 + t_3 \rvert , \lvert t_3 + t_1 \rvert \ge 1
and
|
t
1
+
t
2
+
t
3
|
≥
1
\lvert t_1 + t_2 + t_3 \rvert \ge 1
with
σ
1
,
σ
2
,
σ
3
>
−
K
\sigma _1 , \sigma _2, \sigma _3 > -K
and
σ
1
+
σ
2
,
σ
2
+
σ
3
,
σ
3
+
σ
1
>
1
−
K
\sigma _1 +\sigma _2, \sigma _2 + \sigma _3, \sigma _3 + \sigma _1 > 1-K
, where
K
K
is a positive integer, from bounds for the Hurwitz zeta function which are shown by Bourgain’s bounds for exponential sums.