Mixed sums of infinite multiple zeta-star values with restricted indices

Abstract

We study arithmetic sums of infinite multiple zeta-star values subject to upper or lower bounds on their indices. Let $η(\mathcal D_q)$ be the image, under the zeta-star correspondence, of the admissible infinite index sequences with all indices at most $q$, and let $\overline{η(\mathcal T_p)}$ be the closure of the image of the sequences with all indices at least $p$. We prove that $η(\mathcal D_q)+\overline{η(\mathcal T_p)}$ is an interval exactly when $p\leq q$; in that case it is $[ζ^\star(\{q\}^{\infty})+1,\infty)$. We also show that sums of two upper-bounded sets are half-lines, even with different bounds, whereas sums of two lower-bounded sets are never intervals. The proofs combine a uniform product estimate with natural gap constructions. Lateral thickness handles the diagonal case $p=q\geq3$, while a non-strict local bridge--gap criterion treats $p=q=2$ directly. Compact truncations and explicit omitted intervals yield the remaining assertions.