On a conjecture of Shanks

The conjecture in question concerns the function ϕnϕn related to the distribution of the zeroes of the Riemann zeta-function, γnγn, over the Gram points gngn. It is the purpose of this article to show that for any α>0α>0 the sum∑n=1KϕnKα→0, and this was conjectured, on numerical evidence, by Shanks (1961) [7] to be true for α=12.