Conjecture on inter-trivial zeros of multiple zeta-functions

Let $r\geq 1$ , and let an inter-trivial zero (ITZ) of $\zeta_r(s)$ mean a real zero on the negative real line other than the known zeros at the negative even integers. For any $n\in\mathbb{N}$ , consider the interval $(-2n,-2(n-1))$ . Inter-trivial zero conjecture. There are $(r-1)$ inter-trivial zeros of $\zeta_r(s)$ on the interval

(-2n,-2(n-1))

for any $n\in\mathbb{N}$ . Numerical computations suggest this pattern, which would imply that more inter-trivial zeros appear as $r$ increases; no proof is given in the source.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Conjecture on inter-trivial zeros of multiple zeta-functions
Let $r\geq 1$ and $n\in\mathbb{N}$ . An inter-trivial zero (ITZ) is a real zero of $\zeta_r(s)$ lying between consecutive trivial zeros on the negative real axis. Inter-trivial zero conjecture. There are exactly $r-1$ ITZs of $\zeta_r(s)$ in the interval
$(-2n,-2(n-1))$
for every $n\in\mathbb{N}$ . If true, this predicts that the number of inter-trivial zeros increases with the depth $r$ . The claim is based on numerical computations; no proof or disproof is given in the source.
source: Kohji Matsumoto and Ilija Tanackov, “On the behavior of multiple zeta-functions with identical arguments on the real line I”, arXiv:2012.01712 (2020).

Sources & referencesView supporting material

Primary source

Kohji Matsumoto, Toshiki Matsusaka and Ilija Tanackov, “On the behavior of multiple zeta-functions with identical arguments on the real line”, arXiv:2012.01720 (2020).

1 week ago

Conjecture on inter-trivial zeros of multiple zeta-functions

Equivalent formulations 1

Conjecture on inter-trivial zeros of multiple zeta-functions

Primary source

Progress summary

Solutions 0