Nested Sums of Symbols and Renormalized Multiple Zeta Values

We define renormalized nested sums of symbols on ℝ which satisfy stuffle relations. For appropriate symbols, these give rise to renormalized Euler-Zagier-Hoffman multiple zeta (and Hurwitz zeta) functions which satisfy stuffle relations at all arguments. We show the rationality of renormalized multiple zeta values at non-positive integer arguments. These results generalize to radial symbols on ℝⁿ giving rise to a higher-dimensional analog of multiple zeta functions.