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. They are named after
Derrick Henry Lehmer
Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991), almost always cited as D.H. Lehmer, was an American mathematician significant to the development of computational number theory. Lehmer refined Édouard Lucas' work in the 1930s and ...
, who discovered the pair of zeros
:
(the 6709th and 6710th zeros of the zeta function).
More precisely, a Lehmer pair can be defined as having the property that their complex coordinates
and
obey the inequality
:
for a constant
.
It is an unsolved problem whether there exist infinitely many Lehmer pairs.
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.
See also
*
Montgomery's pair correlation conjecture
In mathematics, Montgomery's pair correlation conjecture is a conjecture made by that the pair correlation between pairs of zeros of the Riemann zeta function (normalized to have unit average spacing) is
:1-\left(\frac\right)^ + \delta(u),
which ...
References
{{reflist, refs=
[{{citation
, last1 = Csordas , first1 = George
, last2 = Smith , first2 = Wayne
, last3 = Varga , first3 = Richard S. , author3-link = Richard S. Varga
, doi = 10.1007/BF01205170
, issue = 1
, journal = ]Constructive Approximation
''Constructive Approximation'' is "an international mathematics journal dedicated to Approximations and Expansions and related research in computation, function theory, functional analysis, interpolation spaces and interpolation of operators, nume ...
, mr = 1260363
, pages = 107–129
, title = Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann hypothesis
, volume = 10
, year = 1994
[{{citation
, last = Lehmer , first = D. H. , authorlink = Derrick Henry Lehmer
, doi = 10.1007/BF02401102
, journal = Acta Mathematica
, mr = 0086082
, pages = 291–298
, title = On the roots of the Riemann zeta-function
, volume = 95
, year = 1956, doi-access = free
]
[{{citation, last1=Rodgers , first1=Brad , last2=Tao , first2=Terence, author2-link=Terence Tao , arxiv=1801.05914 , title=The De Bruijn–Newman constant is non-negative , orig-year=2018, bibcode=2018arXiv180105914R , mr = 4089393 , doi = 10.1017/fmp.2020.6, journal = Forum Math. Pi, volume = 8, year = 2020]
[{{citation, work=What's New, title=Lehmer pairs and GUE, date=January 20, 2018, first=Terence, last=Tao, authorlink=Terence Tao, url=https://terrytao.wordpress.com/2018/01/20/lehmer-pairs-and-gue/]
Analytic number theory