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Lehmer's Phenomenon


LehmersPhenomenon

The appearance of nontrivial zeros (i.e., those along the critical strip with R[z]=1/2) of the Riemann zeta function zeta(z) very close together. An example is the pair of zeros zeta(1/2+(7005+t)i) given by t_1 approx 0.06286617... and t_2 approx 0.1005646..., illustrated above in the plot of |zeta(1/2+(7005+t)i)|^2. This corresponds to the region near Gram point g_(6707.6) (Lehmer 1956; Edwards 2001, p. 178).

Let t_n be the nth nontrivial root of zeta(1/2+it), and consider the local extrema of zeta(1/2+it). Then the values of n after which the absolute value of the local extremum between t_n and t_(n+1) decreases are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, ... (OEIS A114886).


See also

Critical Strip, Gram's Law, Riemann Zeta Function, Riemann Zeta Function Zeros

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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.Edwards, H. M. "Lehmer's Phenomenon." §8.3 in Riemann's Zeta Function. New York: Dover, pp. 175-179, 2001.Lehmer, D. H. "On the Roots of the Riemann Zeta-Function." Acta Math. 95, 291-298, 1956.Sloane, N. J. A. Sequence A114886 in "The On-Line Encyclopedia of Integer Sequences."Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 357-358, 1991.

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Lehmer's Phenomenon

Cite this as:

Weisstein, Eric W. "Lehmer's Phenomenon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LehmersPhenomenon.html

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