Asymptotic Estimation of Ξ(1/2): on a Conjecture of Farmer and Rhoades

We verify a very recent conjecture of Farmer and Rhoades on the asymptotic rate of growth of the derivatives of the Riemann xi function at s = 1/2. We give two separate proofs of this result, with the more general method not restricted to s = 1/2. We briefly describe other approaches to our results, give a heuristic argument, and mention supporting numerical evidence. Introduction Let ξ be the Riemann xi function, given by ξ(s) = 12s(s − 1)π−s/2Γ(s/2)ζ(s), where Γ is the Gamma function and ζ is the Riemann zeta function [7, 17]. It satisfies the functional equation ξ(s) = ξ(1 − s) and is entire of order 1. The functional equation implies that all odd order derivatives of ξ vanish at s = 1/2. On the other hand, estimation of the even order derivatives there has been an open problem and of interest from many points of view. Our main results are Propositions 1 and 2. Proposition 1. Let ξ be the Riemann xi function and n a positive integer. Then as n → ∞ we have (1) ln ξ ( 1 2 ) = 2n ln(lnn)− 2 ( ln 2 + 1 lnn ) n+ 9 4 ln(2n)− 3 4 ln(lnn) +O(1). Proposition 2. For real s and j → ∞ we have ξ(s) = j(j − 1) 2j−1 (j − 2)(s−1)/2 ln(j − 2) { 1 + (−1) [ ln(j − 2) j − 2 ]s−1/2} × [ ln ( j − 2 π ) − ln ( ln ( j − 2 π )) + o(1) ]j−3/2 exp [ − (j − 2) ln(j − 2) ] . (2) Proposition 1 is in response to a conjecture of Farmer and Rhoades [8] that ln ξ(1/2) should increase very regularly and not too much faster than linearly as n → ∞. They made this conjecture in the course of a study of the effect of repeated differentiation upon the zero spacings of a real entire function of order 1. Current limited numerical evidence due to Kreminski [15] supports the conjecture and equation (1). The conjecture of Farmer and Rhoades was based upon the zero count n±(r) of the function Ξ(z) = ξ(1/2 + iz) in the intervals (0, r] and [−r, 0), Received by the editor March 17, 2006 and, in revised form, April 24, 2008. 2000 Mathematics Subject Classification. Primary 11M06, 30D15.