| Summary: | 碩士 === 國立中正大學 === 應用數學研究所 === 103 === The classical Euler decomposition expresses a product of two Riemann zeta values as double Euler sums and it leads to a weighted sum
formula among double Euler sums.
Through a particular integral representation of Riemann's zeta values, we are able to carry out the shuffle product of $n$ Riemann zeta values.
As results, we produce some weighted sum formulas among multiple zeta values of depth 2, 3 and 4. In particular when the depth $n=4$,
the weighted sum formula is given by
\begin{align*}
&\sum_{|\balpha|=k+7}\zeta(\alpha _{1},\alpha_2,\alpha_3,\alpha _{4}+1)\left \{ 2^{\alpha _{2}-1}3^{\alpha _{3}-1}\left ( 4^{\alpha _{4}}-3^{\alpha _{4}} \right )
\right.\\
&\qquad \quad\quad\quad\quad \left. -\left ( 2^{\alpha _{2}+\alpha _{3}-2}+2^{\alpha _{3}-1} \right )
\left ( 3^{\alpha _{4}}-2^{\alpha _{4}} \right )+\left (2^{\alpha _{4}} -1\right )\right \}\\
&-\sum_{|\balpha|=k+6}\zeta(1,\alpha_2,\alpha_3,\alpha _{4}+1)\left \{ 2^{\alpha _{3}-1}\left ( 3^{\alpha _{4}}-2^{\alpha _{4}} \right )-\left (2^{\alpha _{4}} -1\right )\right \}\\
& +\sum_{|\balpha|=k+5}\zeta(1,1,\alpha_3,\alpha_4+1)\left \{ 2^{\alpha _{4}}-1 \right \}-\zeta \left ( 1,1,1,k+5 \right)\\
& =\frac{1}{24}\sum_{\left | \mathbf{a} \right |=k}\zeta(a_{1}+2)\zeta(a_{2}+2)\zeta(a_{3}+2)\zeta(a_{4}+2).
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