Derivatives and fast evaluation of the Tornheim zeta function

作者：Borwein Jonathan M; Dilcher Karl^*

来源：Ramanujan Journal, 2018, 45(2): 413-432.

DOI：10.1007/s11139-017-9890-9

• Tornheim zeta function
• Mordell-Tornheim double series
• Incomplete gamma function
• Polylogarithm
• Bernoulli polynomials

We study analytic properties of the Tornheim zeta function , which is also named after Mordell and Witten. In particular, we evaluate the function () at and, as our main result, find the derivative of this function at . Our principal tool is an identity due to Crandall that involves a free parameter and provides an analytic continuation. Furthermore, we derive special values of a permutation sum. Throughout this paper, we show by way of examples that Crandall's identity can be used for efficient and high-precision evaluations of the Tornheim zeta function.

• 出版日期2018-2