On a Conjecture of Chowla and Milnor

Abstract

Abstract. In this paper, we investigate a conjecture due to S. and P. Chowla and its generalization by Milnor. These are related to the delicate question of non-vanishing of L-functions associated to periodic functions at integers greater than 1. We report on some progress in relation to these conjectures. In a different vein, we link them to a conjecture of Zagier on multiple zeta values and also to linear independence of polylogarithms.

A Number Field Extension of a Question of Milnor

“…Secondly, typically one is interested in irrationality of ζ(2d+1)/π 2d+1 as well as that of ζ(2d+1) and this generalisation predicts the irrationality of both these numbers. Following is this extension suggested by the authors (see [3]):…”

“…In such cases, the mathematics is somewhat amenable and one can derive similar lower bounds for these dimensions as has been done in the earlier works [3] and [5].…”

“…They also obtained a conditional improvement of this lower bound and noted that any unconditional improvement of this "half" threshold will have remarkable consequences in relation to irrationality of the numbers ζ(2d + 1)/π 2d+1 . Furthermore in [3], the authors suggested a generalisation of the original conjecture of Milnor. There are at least two reasons for considering such a generalisation.…”

“…In [3], the second and the third author with M. Ram Murty studied Milnor's conjecture and derived a non-trivial lower bound for the dimension of V (k, Q), namely that the dimension is at least half of the conjectured dimension. They also obtained a conditional improvement of this lower bound and noted that any unconditional improvement of this "half" threshold will have remarkable consequences in relation to irrationality of the numbers ζ(2d + 1)/π 2d+1 .…”

“…Milnor [7] formulated a conjecture about rational linear independence of some special Hurwitz zeta values. In [3], this conjecture was studied and an extension of Milnor's conjecture was suggested. In this note, we investigate the number field generalisation of this extended Milnor conjecture.…”