The Strong Chowla-Milnor spaces and a conjecture of Gun, Murty and Rath

TL;DR

This paper establishes lower bounds for the dimensions of Strong Chowla-Milnor spaces and generalized Zagier spaces, linking these bounds to conjectures on the irrationality of zeta values and advancing understanding of multiple zeta values over number fields.

Contribution

It proves a non-trivial lower bound for the dimension of Strong Chowla-Milnor spaces and relates these bounds to conjectures on zeta values, also defining and analyzing generalized Zagier spaces.

Findings

Lower bound for the dimension of Strong Chowla-Milnor spaces

Conditional improvement of the lower bound based on conjectures

Dimension of generalized Zagier spaces V_{4d+2}(K) is at least 2

Abstract

In a recent work, Gun, Murty and Rath formulated the Strong Chowla-Milnor conjecture and defined the Strong Chowla-Milnor space. In this paper, we prove a non-trivial lower bound for the dimension of these spaces. We also obtain a conditional improvement of this lower bound and noted that an unconditional improvement of this lower bound will lead to irrationality of both $\zeta(k)$ and $\zeta(k)/ \pi^k$ for all odd positive integers $k>1$. Following Gun, Murty and Rath, we define generalized Zagier spaces $V_p(K)$ for multiple zeta values over a number field $K$. We prove that the dimension of $V_{4d+2}(K)$ for $d\geq 1$, is at least 2, assuming a conjecture of Gun, Murty and Rath.