On the dimension of Chowla-Milnor space

TL;DR

This paper provides an alternative proof for a conditional lower bound on the dimension of Chowla-Milnor spaces, which are linked to the irrationality of certain zeta function ratios at odd integers.

Contribution

It offers a new proof of a key theorem regarding Chowla-Milnor spaces, potentially impacting the understanding of zeta function irrationality.

Findings

Alternative proof of the conditional lower bound for Chowla-Milnor spaces

Reinforces the connection between these spaces and the irrationality of zeta ratios

Highlights the importance of unconditional bounds for future research

Abstract

In a recent work, Gun, Murty and Rath defined the Chowla-Milnor space and proved a non-trivial lower bound for these spaces. They also obtained a conditional improvement of this lower bound and noted that an unconditional improvement of their lower bound will lead to irrationality of $\zeta(k)/ \pi^k$ for odd positive integers $k>1$. In this paper, we give an alternate proof of their theorem about the conditional lower bound.