Abstract

We nd and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apery sums). The study of non-alternating sums leads to an investigation of dierent types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experimental results involving polylogarithms in the golden ratio. AMS (1991) subject classication: Primary 40B05, 33E20, Secondary 11M99, 11Y99. Key words: binomial sums, multiple zeta values, log-sine integrals, Clausens function, multiple Clausen values, polylogarithms, Apery sums. 1 1 Introduction We shall begin by studying the central binomial sum S(k), given as: S(k) := 1 X n=1 1 n k 2n n (1) for integer k. A classical evaluation is S(4) = 17 36 (4). Using a mixture of integer relation and other computational techniques, we uncover remarkable links to values multi-dimensional polylogarithms of sixth roots of unity which we call multip...

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