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Pair of zeros of the Riemann zeta function
In the study of the
Riemann hypothesis , a Lehmer pair is a pair of
zeros of the
Riemann zeta function that are unusually close to each other.
[1] They are named after
Derrick Henry Lehmer , who discovered the pair of zeros
1
2
+
i
7005.06266
…
1
2
+
i
7005.10056
…
{\displaystyle {\begin{aligned}&{\tfrac {1}{2}}+i\,7005.06266\dots \\[4pt]&{\tfrac {1}{2}}+i\,7005.10056\dots \end{aligned}}}
(the 6709th and 6710th zeros of the zeta function).
[2]
Unsolved problem in mathematics :
Are there infinitely many Lehmer pairs?
More precisely, a Lehmer pair can be defined as having the property that their complex coordinates
γ
n
{\displaystyle \gamma _{n}}
and
γ
n
+
1
{\displaystyle \gamma _{n+1}}
obey the inequality
1
(
γ
n
−
γ
n
+
1
)
2
≥
C
∑
m
∉
{
n
,
n
+
1
}
(
1
(
γ
m
−
γ
n
)
2
+
1
(
γ
m
−
γ
n
+
1
)
2
)
{\displaystyle {\frac {1}{(\gamma _{n}-\gamma _{n+1})^{2}}}\geq C\sum _{m\notin \{n,n+1\}}\left({\frac {1}{(\gamma _{m}-\gamma _{n})^{2}}}+{\frac {1}{(\gamma _{m}-\gamma _{n+1})^{2}}}\right)}
for a constant
C
>
5
/
4
{\displaystyle C>5/4}
.
[3]
It is an unsolved problem whether there exist infinitely many Lehmer pairs.
[3]
If so, it would imply that the
De Bruijn–Newman constant is non-negative,
a fact that has been proven unconditionally by Brad Rodgers and
Terence Tao .
[4]
See also
References
^ Csordas, George; Smith, Wayne;
Varga, Richard S. (1994), "Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann hypothesis",
Constructive Approximation , 10 (1): 107–129,
doi :
10.1007/BF01205170 ,
MR
1260363 ,
S2CID
122664556
^
Lehmer, D. H. (1956), "On the roots of the Riemann zeta-function",
Acta Mathematica , 95 : 291–298,
doi :
10.1007/BF02401102 ,
MR
0086082
^
a
b
Tao, Terence (January 20, 2018),
"Lehmer pairs and GUE" , What's New
^ Rodgers, Brad;
Tao, Terence (2020) [2018], "The De Bruijn–Newman constant is non-negative",
Forum Math. Pi , 8 ,
arXiv :
1801.05914 ,
Bibcode :
2018arXiv180105914R ,
doi :
10.1017/fmp.2020.6 ,
MR
4089393 ,
S2CID
119140820