Bounds for the Tornheim double zeta function

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.