Lehmer's phenomenon is the appearance of unusually close nontrivial zeros of the Riemann zeta function on the critical
line.
An example is the pair of zeros given by and , illustrated above in the plot of . This lies between
Gram points and (Lehmer 1956; Edwards 2001, p. 178).
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. Analysis1, 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 -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.
on the critical
line ![Re[z]=1/2](/images/equations/LehmersPhenomenon/Inline2.svg)
.
An example is the pair of zeros 
given by 
and 
, illustrated above in the plot of 
. This lies between
Gram points 
and 
(Lehmer 1956; Edwards 2001, p. 178).
denote the positive imaginary
parts of the nontrivial zeros on the critical
line, in increasing order, and set 
. For 
, let 
be the maximum of 
between 
and 
, where 
is the xi-function. The positive integers 
for which 
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).
-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