Who's up to the challenge? 75

Calculus Level 5

$\zeta ({ s }_{ 1 },\ldots ,{ s }_{ k })=\sum _{ 0<{ n }_{ 1 }<{ n }_{ 2 }<\cdots <{ n }_{ k } }^{ }{ \frac { 1 }{ { n }_{ 1 }^{ { s }_{ 1 } }{ n }_{ 2 }^{ { s }_{ 2 } }\cdots { n }_{ k }^{ { s }_{ k } } } }$

Define the multiple zeta function as shown above. Then find

$\lim _{ s_{ 2 }\rightarrow 0 }{ \lim _{ { s }_{ 1 }\rightarrow 0 }{ \zeta (s_{ 1 },{ s }_{ 2 }) } } +\lim _{ { s }_{ 1 }\rightarrow 0 }{ \lim _{ { s }_{ 2 }\rightarrow 0 }{ \zeta ({ s }_{ 1 },{ s }_{ 2 }) } } .$

Note :

The multiple zeta function in the limit is defined by its analytic continuation.

The answer is 0.75.

1 solution

Mark Hennings
Jun 21, 2016

The result can be found in this paper .

However, the question should be much clearer that we are looking at the analytic continuation of the multiple zeta function. The formula in the question is only valid for $\mathrm{Re}\,s_k > 1$ and $\mathrm{Re}\,(s_1+s_2+\cdots+s_k) > k$ .

The above-mentioned paper shows that the multiple zeta function $\zeta(s_1,s_2)$ can be extended to a meromorphic function on $\mathbb{C}^2$ , and that $\lim_{s_1 \to 0} \lim_{s_2 \to 0} \zeta(s_1,s_2) \; = \; \tfrac13 \qquad \qquad \lim_{s_2 \to 0} \lim_{s_1 \to 0} \zeta(s_1,s_2) \; = \; \tfrac{5}{12}$ The sum of these limits is $\boxed{\tfrac34}$ .

The oddness of this result only confirms that asking for solutions involving multidimensional analytic extensions is a bit extreme.

Who's up to the challenge? 75

The answer is 0.75.

1 solution

0 pending reports