Notes on the Rieman ζ-function, 3

Let rho(t) denote the fractional part of t. H the Hilbert space L-2(0, + infinity), B the subspace, of H of functions f(t) = Sigma(k=1)(n) c(k)p(0(k)/t), where n is an element of N, c(k) is an element of C, and 0 < theta(k) less than or equal to 1 for 1 less than or equal to k less than or equal to n, chi the characteristic function of ]0, 1] and D(lambda) the distance in H between chi and B-lambda, the subspace of B of functions f such that all theta(k) greater than or equal to lambda A well-known result of B. Nyman and A. Beurling implies that the Riemann hypotheses is equivalent to the statement lim(lambda-->0) D(lambda)= 0. We prove here that inf(0<lambda<1) D(lambda) root log(2/lambda) > 0, and we conjecture that lim(lambda-->0) D(lambda) root log(1/lambda) =root 2+gamma-log(4 pi), where gamma denotes Euler's constant. This conjecture is supported by numerical experiments. (C) 2000 Academic Press.