Does this sum equal zeta(3)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T09:50:56Z http://mathoverflow.net/feeds/question/55141 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/55141/does-this-sum-equal-zeta3 Does this sum equal zeta(3)? David Speyer 2011-02-11T16:57:28Z 2011-02-11T17:55:48Z <p>Does: <code>$$\sum_{1 \leq i&lt;j} \frac{1}{i j^2} = \sum_{1 \leq k} \frac{1}{k^3}?$$</code></p> <p>Motivation: Call the above sum $S$, and let <code>$$T := \sum_{ GCD(i,j)=1} \frac{1}{\max(i,j) i j}.$$</code> The sum $T$ came up in a computation on Jim Propp's question <a href="http://mathoverflow.net/questions/54731/sums-of-fractional-parts-of-linear-functions-of-n" rel="nofollow">here</a>. Numerical computation suggested that $T$ is extremely close to $3$.</p> <p>It is not hard to show that <code>$$T = \zeta(3)^{-1} \sum \frac{1}{\max(i,j) i j} = \zeta(3)^{-1} \left( \sum_{k} \frac{1}{k^3} + 2 \sum_{i&lt;j} \frac{1}{i j^2} \right) = 1 + 2 \zeta(3)^{-1} S,$$</code> by breaking into cases according to whether <code>$i&lt;j$</code>, <code>$i=j$</code> or <code>$i&gt;j$</code>. So $T=3$ iff $S=\zeta(3)$.</p> <p>As I describe in the above linked thread, numerical computations suggest that the sums agree to $20$ digits of accuracy. What is going on?</p> http://mathoverflow.net/questions/55141/does-this-sum-equal-zeta3/55144#55144 Answer by Marty for Does this sum equal zeta(3)? Marty 2011-02-11T17:05:36Z 2011-02-11T17:05:36Z <p>Hi David,</p> <p>This is the first example of a multiple zeta identity. Your sum S is just $\zeta(1,2)$, where the multiple zeta value is defined by: $$\zeta(s_1, s_2, \ldots, s_k) = \sum_{0 &lt; n_1 &lt; n_2 &lt; \cdots n_k} \left( \prod_{i=1}^k n_i^{-s_i} \right).$$</p> <p>Your identity $\zeta(1,2) = \zeta(3)$ was discovered by Euler <a href="http://en.wikipedia.org/wiki/Multiple_zeta_function" rel="nofollow">according to Wikipedia</a>. </p>