THE STRONG CHOWLA–MILNOR SPACES AND A CONJECTURE OF GUN, MURTY AND RATH

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 proved a non-trivial lower bound for the dimension of these spaces. We also obtained a conditional improvement of this lower bound and noted that an unconditional improvement of this lower bound will lead to irrationality of both ζ(k) and ζ(k)/πk for all odd positive integers k. Following Gun, Murty and Rath, we define generalized Zagier spaces Vp(K) for multiple zeta values over a number field K. We prove that the dimension of V4d+2(K) for d ≥ 1, is at least 2, assuming a conjecture of Gun, Murty and Rath.