The appearance of nontrivial zeros (i.e., those along the critical strip with )
of the Riemann zeta function very close together. An example is the pair of zeros
given by
and ,
illustrated above in the plot of . This corresponds to the region near
Gram point (Lehmer 1956; Edwards 2001, p. 178).
Let
be the th
nontrivial root of ,
and consider the local extrema of . Then the values of after which the absolute value of the local extremum between
and
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).
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.
![R[z]=1/2](/images/equations/LehmersPhenomenon/Inline1.svg)
)
of the Riemann zeta function 
very close together. An example is the pair of zeros
given by 
and 
,
illustrated above in the plot of 
. This corresponds to the region near
Gram point 
(Lehmer 1956; Edwards 2001, p. 178).
be the 
th
nontrivial root of 
,
and consider the local extrema of 
. Then the values of 
after which the absolute value of the local extremum between
and 
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).
-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