Lehmer's Phenomenon

D. H. Lehmer (Lehmer, D. H. "On the Roots of the Riemann Zeta-Function." Acta Math. 95, 291-298, 1956) discovered  cases where the Riemann zeta function has zeros that are "only just" on the critical line: two zeros of the zeta function are so close together that it is unusually difficult to find a sign change between them. This is called "Lehmer's phenomenon", and first occurs at the zeros with imaginary parts 7005.063 and 7005.101, which differ by only .04 while the average gap between other zeros near this point is about 1. The discovery of non-simple roots, or the discovery of a local positive minimum, or a local negative maximum,  implies a violation of the Riemann hypothesis. The relation between Lehmer's phenomenon and the Riemann hypothesis can be quantitatively expressed through the de Bruijn-Newman constant, which we describe below. We have applied Machine Learning to relate the occurrence of Lehmer pairs to the behavior of the zeta function at Gram points. 

The Riemann ξ-function can be expressed in the form

where

The Riemann Hypothesis is the statement that all zeros of ξ are real. In the literature Ht  is defined as

H0 and the Riemann ξ-function are related through H0(x) = ξ(x/2)/8,  so the Riemann Hypothesis is also equivalent to the statement that all zeros of H0 are real.

In 1950, De Bruijn (N. C. de Bruijn The roots of trigonometric integrals, Duke J. Math 17 (1950), pp. 197-226.)     showed that Ht has only real zeros for t > 1/2, and if Ht has only real zeros for some t, then that property holds for all larger values of t.  C. M. Newman (C. M. Newman, Fourier transforms with only real zeros, Proc. Amer. Math. Soc. 61 (1976),

pp. 245-251) further showed that there is a real constant Λ (−∞ < Λ ≤ 1/2) such that Ht has only real zeros if and only if t  ≥ Λ.  Λ is now called the de Bruijn-Newman constant. The Riemann Hypothesis is equivalent to the conjecture that Λ ≤ 0. On the other hand, C. M. Newman conjectured that Λ ≥ 0. If Newman’s conjecture is true, then the Riemann Hypothesis, even if it is true, is only barely so, as even slight perturbations of the zeta function give rise to zeros that are not on the critical line. There has been extensive research activity in finding lower bounds for Λ (G. CSORDAS, A.M. ODLYZKO W. SMITH, , AND R. S. VARGA, "A NEW LEHMER PAIR OF ZEROS AND A NEW LOWER BOUND FOR THE DE BRUIJN-NEWMAN CONSTANT Λ"   Electronic Transactions on Numerical Analysis. Volume 1, pp. 104-111, December 1993;  G. Csordas, W. Smith, and R. S. Varga, "Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann Hypothesis", Constr. Approx., 1994).  

It is convenient to order the zeros of H0, {xj(0)}, j=1, ∞,  in Re z > 0 according to increasing modulus, and, from the evenness of H0 it follows that x-j = xj.  The references mentioned earlier derived limits on the de Bruijn-Newman constant by considering close pairs of Riemann zeta zeros.   With k a positive integer, let xk(0) and xk+1(0) (with 0 <  xk(0) < xk+1(0)) be two consecutive simple positive zeros of H0, and set  ∆k := xk+1(0) − xk(0).

Then, {xk(0), xk+1(0)} is a Lehmer pair of zeros of H0 if  ∆k 2 · g k ( 0 ) < 4 / 5 ,  where 

Given a Lehmer pair of zeros, we have the Theorem:

Let {xk(0), xk+1(0)} be a Lehmer pair of zeros of H0 . If gk(0) ≤ 0, then Λ > 0. If gk(0) > 0, set

so that −1/[8gk(0)] < λk < 0. Then, the de Bruijn-Newman constant Λ satisfies  λk ≤ Λ. 

The best limit on from considering Lehmer zero pairs currently is -1.14541 x 10-11 (from Yannick Saouter, Xavier Gourdon, Patrick Demichel: An improved lower bound for the de Bruijn-Newman constant. Math. Comput. 80(276): 2281-2287 (2011)). This comes from the pair of zeros 7954022502373.43289015385 and 79540225023703.43289494012.  

Critical Behaviour for Riemann zeta function? Sharp Transition