Lehmer's Phenomenon

The appearance of nontrivial zeros (i.e., those along the Critical Strip with $\Re[z]=1/2$) of the Riemann Zeta Function $\zeta(z)$ very close together. An example is the pair of zeros $\zeta({\textstyle{1\over 2}}+(7005+t)i)$ given by $t_1\approx 0.0606918$ and $t_2\approx 0.100055$, illustrated above in the plot of $\vert\zeta({\textstyle{1\over 2}}+(7005+t)i)\vert^2$.

See also Critical Strip, Riemann Zeta Function

References

Csordas, G.; Odlyzko, A. M.; Smith, W.; and Varga, R. S. ``A New Lehmer Pair of Zeros and a New Lower Bound for the de Bruijn-Newman Constant.'' Elec. Trans. Numer. Analysis 1, 104-111, 1993.

Csordas, G.; Smith, W.; and Varga, R. S. ``Lehmer Pairs of Zeros, the de Bruijn-Newman Constant and the Riemann Hypothesis.'' Constr. Approx. 10, 107-129, 1994.

Csordas, G.; Smith, W.; and Varga, R. S. ``Lehmer Pairs of Zeros and the Riemann $\xi$-Function.'' In Mathematics of Computation 1943-1993: A Half-Century of Computational Mathematics (Vancouver, BC, 1993). Proc. Sympos. Appl. Math. 48, 553-556, 1994.

Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 357-358, 1991.